# Validation of a Fast and Accurate Magnetic Tracker Operating in the Environmental Field

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## Abstract

**:**

## 1. Introduction

## 2. Setup

- Subtract the individual sensor offsets;
- Make the responses isotropic;
- Refer to a unique co-ordinate system, oriented in accordance with that of a sensor selected as the reference one;

## 3. Model and Algorithms

#### 3.1. Field Model

#### 3.2. Limits

#### 3.3. Best-Fit Degrees-of-Freedom

#### 3.4. Scalability

## 4. Trajectory Performance

- accuracy of the calibration and subsequent consistency of the magnetometric data
- accuracy of knowledge of the sensor positions
- presence of time-dependent noise or other disturbances preventing accurate measurements
- presence of relevant inhomogeneities of the ambient field.

- intensity of the target dipole $|\overrightarrow{m}|$;
- distance of the target from the sensor array
- position and orientation of the target with respect the sensor array (the performance is not isotropic around the array center, and the anisotropy depends also on the dipole orientation)

#### 4.1. Circle

#### 4.2. Helix

#### 4.3. Tuning the 8-Parameter Fit

## 5. Time Performance

## 6. Guess Criticality

- the initial guess of $\overrightarrow{m}$ is not critical (just assign reasonable values);
- the initial guess of ${\overrightarrow{B}}_{g}$ is not critical (just assign reasonable values);
- the initial guess of $\overrightarrow{r}$ is critical, but whenever it is known that the target is on one side of the array (a given sign of z), and is not too displaced from the array axis, an axial guess for $(x,y)$ together with a reasonable guess for z will work;
- the larger is the z guess, the less critical is the $(x,y)$ choice, but large values make the convergence slower, and too large values will prevent convergence;
- in case of correct guess, the iteration number depends weakly of the termination conditions and on the selected guess;
- good guesses bring to convergence within few tens of iterations;
- bad guesses with too large $\left|r\right|$ bring to (wrong) convergence within a few iteration steps;
- some wrong guesses with more reasonable $\left|r\right|$ bring to (wrong) convergence quite slowly, with many iteration steps (setting a limited number of iteration steps will help avoid wasting time with useless calculations);
- wrong convergence is easily detected, because the (local) minimum found is orders of magnitude larger than the absolute one;
- in case of a wrong convergence detected, trying other initial guesses having different $\overrightarrow{r}$ will help;
- in the considered application of eye-tracking, the limited RoI size makes a certainly-good guess possible;
- the accuracy in reconstructing trajectories is generally good, but the best performance is obtained when the target moves on a surface nearly parallel to $\overrightarrow{m}$: in eye-tracking application, a radial orientation of $\overrightarrow{m}$ will be a favorite choice.

## 7. Conclusions

## 8. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The sensor array contains 8 sensors distributed on two parallel PCBs (three of them, highlighted with blue circles are a $z=0$ and other five (orange circles) at $z=16.6$ mm). The system is designed for eye-tracking purposes. A 20 mm diameter cylinder is used to emulate eye motion. It hosts the magnetic source (target), which performs either circular or helix trajectories.

**Figure 2.**The field components are referred to co-ordinates defined within the reference sensor chip ${S}_{ref}$ (in black), which might be slightly misaligned with respect to the PCB frame (in red). This can lead to inaccuracy in the position of the other chips (${S}_{k}$). In addition, each chip contains three sensors (represented by colored dots), which are (submillimetrically) displaced with respect to each other.

**Figure 3.**Projections on the $xy$ e $zy$ planes of a 8P reconstructed circular trajectory. The red curves are elliptical best-fits. Residuals as small al 6.1 μm and 92 μm are obtained, respectively. The corresponding 9P analysis lead to results that are similar (5.6 μm) in the $xy$ projection and worse (95 μm) in the $xy$ projection.

**Figure 4.**3D views of reconstructed helix trajectory, the axes units are expressed in mm. The magnet is radially oriented and follows a helix trajectory 10 mm in radius and 1 mm in pitch, around the z axis. Beside the spatial position the tracker retrieves the dipole vector: red-blue dots are used to represent this data. The left tracking is obtained by 9P and suffers of evident (nevertheless submillimetric) distortions of the z co-ordinate. The right tracking uses the same magnetometric data, but a 8P analysis.

**Figure 5.**Tracking of $xy$ projections as obtained with 8P, using for $|\overrightarrow{m}|$ its value resulting from 9P (black) and values increased (blue, green) or decreased (red, magenta) by 20% and 40%, respectively.

**Figure 6.**Average iteration number as a function of the guess accuracy for the 8P (blue) and 9P (red) cases. The cases with q = 1 and q = 670 correspond to using one of the nearest neighbors as an initial guess, while q = 335 corresponds to using diametrically opposite points.

**Figure 7.**2D convergence maps of $xy$ (

**a**,

**b**) and $xz$ (

**c**,

**d**) sections. The remaining 7 parameters are set to their exact values (

**a**,

**c**) or to their default values (

**b**,

**d**). The target (white dot) is in a nearly axial position, at $z=31$ mm. Unexpectedly, using wrong guesses for ${\overrightarrow{B}}_{g}$ and $\overrightarrow{m}$ and wrong (larger) z makes the ($x,y$) convergence area larger than using exact values. All these maps describe a 300 mm × 300 mm area, the yellow circle in (

**c**) corresponds to a sphere (26 mm in diameter) that is the typical size of human eye bulb.

**Figure 8.**2D convergence maps of $xy$ (

**a**,

**b**) and $xz$ (

**c**,

**d**) sections. The remaining 7 parameters are set to their exact values (

**a**,

**c**) or to their default values (

**b**,

**d**). The target (white dot) is in a more peripheral position (larger distance from the array axis), with respect to the case of Figure 7, at $z=31$ mm. It is confirmed that using wrong guesses for ${\overrightarrow{B}}_{g}$ and $\overrightarrow{m}$ and wrong (larger) z makes the ($x,y$) convergence area larger than using the exact values.

**Figure 9.**2D convergence maps of $xy$ sections. The ${\overrightarrow{B}}_{g}$ and $\overrightarrow{m}$ guess is set at the default, and the z guess is variously assigned: from 8 mm (

**a**), to 20 mm (

**b**), 60 mm (

**c**) and 80 mm (

**d**). The $z=$ 40 mm case was already shown in Figure 8b.

**Table 1.**Default values used as starting guesses (unless differently specified) for the target position $\overrightarrow{r}$, the environmental field ${\overrightarrow{B}}_{g}$, and the magnetic dipole $\overrightarrow{m}$.

Quantity | x | y | z |
---|---|---|---|

$\overrightarrow{r}$ | 20 mm | $-20$ mm | 40 mm |

${\overrightarrow{B}}_{g}$ | 20 μT | 20 μT | 20 μT |

$\overrightarrow{m}$ | 600 μAm${}^{2}$ | 600 μAm${}^{2}$ | 600 μAm${}^{2}$ |

**Table 2.**Time performances of 8P and 9P, in the case of next-neighbor guess, in case of diametrically opposite guess and in case of default values assigned programmatically (see Table 1).

Quantity | q = 1 | q = N/2 | Default |
---|---|---|---|

8P (iter. number) | 30 | 55 | 48 |

9P (iter. number) | 20 | 35 | 25 |

8P (ms/tracking) | 7.5 | 13.8 | 12 |

9P (ms/tracking) | 5.4 | 9.5 | 6.8 |

**Table 3.**Co-ordinates of the sensors. The last two lines report the central and peripheral target positions (determined by the tracker) considered in the analyses of this Section.

Object | x (mm) | y (mm) | z (mm) |
---|---|---|---|

sensor 0 | 0 | 0 | 0 |

sensor 1 | 0 | $-27.05$ | 0 |

sensor 2 | 40.64 | $-19.28$ | 0 |

sensor 3 | 0 | $-19.28$ | 16.6 |

sensor 4 | 20.26 | $-45.74$ | 16.6 |

sensor 5 | 40.64 | $-32$ | 16.6 |

sensor 6 | 40.64 | $-6.6$ | 16.6 |

sensor 7 | 20.33 | 7.25 | 16.6 |

${\overrightarrow{r}}_{c}$ | 13 | $-20$ | 30.5 |

${\overrightarrow{r}}_{p}$ | 33 | $-19.8$ | 29.7 |

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**MDPI and ACS Style**

Biancalana, V.; Cecchi, R.; Chessa, P.; Mandalà, M.; Bevilacqua, G.; Dancheva, Y.; Vigilante, A.
Validation of a Fast and Accurate Magnetic Tracker Operating in the Environmental Field. *Instruments* **2021**, *5*, 11.
https://doi.org/10.3390/instruments5010011

**AMA Style**

Biancalana V, Cecchi R, Chessa P, Mandalà M, Bevilacqua G, Dancheva Y, Vigilante A.
Validation of a Fast and Accurate Magnetic Tracker Operating in the Environmental Field. *Instruments*. 2021; 5(1):11.
https://doi.org/10.3390/instruments5010011

**Chicago/Turabian Style**

Biancalana, Valerio, Roberto Cecchi, Piero Chessa, Marco Mandalà, Giuseppe Bevilacqua, Yordanka Dancheva, and Antonio Vigilante.
2021. "Validation of a Fast and Accurate Magnetic Tracker Operating in the Environmental Field" *Instruments* 5, no. 1: 11.
https://doi.org/10.3390/instruments5010011