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HAL
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2D Magnetic Localization Newton-Raphson-Based
Method Considering Measurement Uncertainties for
Smart Surface Applications

Gabriel Géron, Jérémy Terrien, Muneeb Ullah Khan, Hani Al Hajjar,
Christine Prelle

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To cite this version:

Gabriel Géron, Jérémy Terrien, Muneeb Ullah Khan, Hani Al Hajjar, Christine Prelle. 2D Magnetic Localization Newton-Raphson-Based Method Considering Measurement Uncertainties for Smart Surface Applications. IEEE Transactions on Instrumentation and Measurement, 2025, 74, pp.1-11. 10.1109/TIM.2025.3568961. hal-05086016

HAL Id: hal-05086016

https://utc.hal.science/hal-05086016v1

Submitted on 27 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# 2D Magnetic Localization Newton-Raphson-Based Method Considering Measurement Uncertainties for Smart Surface Applications

Gabriel Géron, Jérémy Terrien, Muneeb Ullah Khan, Hani Al Hajjar and Christine Prelle

***Abstract***—Magnetic localization techniques present some interests for micro robotics applications using smart surfaces due to their inherent advantages, such as integration ability or low cost. However, at microscale, these methods often encounter challenges to perform with high precision in real scenario, because of the limitation of calculation algorithms (numerical stability, convergence, approximation methods) and the impact of measurement uncertainties (misalignment, sensor performance, etc.). This paper presents a new 2D magnetic localization method based on a 3D Hall effect sensor array system, in order to localize a cubic magnet. The proposed approach combines an efficient numerical resolution algorithm (Newton-Raphson) with precise analytical models of the magnetic field of the magnet to brave these measurement uncertainties and to achieve high-resolution 2D magnetic localization. Through performance assessments from simulation calculations and experimentations, the operation and efficiency of the proposed method is demonstrated. Results indicate good performance of the Newton-Raphson method with micrometric accuracy, even when there are measurement uncertainties in real scenario. Experimental test realized with a prototype of 3D Hall effect sensor array confirms the efficiency of the 2D magnetic localization with an averaged 12.73 µm localization error. A comparative analysis shows that this new magnetic localization method is well placed in the hierarchy of existing methods, that estimate the 2D position of a magnet by using a Hall effect sensor array system.

***Index Terms***—2D magnetic localization, 3D Hall effect sensor array, analytical magnetic models, calibration process, Newton-Raphson algorithm, permanent magnet, smart surface

## I. INTRODUCTION

All the positioning systems used in micro robotics, the development of smart surface has strongly expanded in order to move and control micro-objects over a relative long distance. The smart surfaces are often described as planar platforms composed of an actuation surface, a sensing component and a control system [1]. The actuation surfaces are generally based on distributed actuation elements that compose a matrix of active independent cells able to generate long-stroke planar motions. In the literature, these actuation surfaces can be realized by using different physical principles as pneumatic [2]-[4], electrostatic [5] [6],

electrothermal [7] [8] or electromagnetic [9]-[11]. For 2D micromanipulation applications like conveying or sorting, an open loop control can be sufficient to perform these tasks, thus requiring no means of measurement to localize micro-objects over the actuation surface. However, this latter has limits when an accurate control of the trajectory is needed to carry out precise manipulation or positioning tasks [12]. In this scenario, a high-resolution position measurement system becomes an essential element to achieve a closed-loop control of the displacement or the positioning of small parts at micro scale.

For micro scale applications, the vision-based system is often employed to localize small objects to ensure the closed loop control. High resolution images of the target shape are captured at high frequency by using high resolution cameras [13]-[15] or with microscopes [16]-[18]. These images are processed to extract information about the position, the moving speed and the pose of the object. These vision-based systems have the benefits to provide high resolution measurement (i.e. sub-micrometric) and allow the control of the target without the need of an added measuring element on the object. However, due to the field of view, they are often limited to small workspace (< 100 mm²), to achieve high measurement resolution. Widening the field of view to measure the larger workspace leads to degradation of the measurement resolution. Consequently, it is very difficult to combine wide-range and high-resolution measurements with the vision-based techniques. To overcome this limitation, numerous solutions have been studied regarding localization techniques. These solutions are based on implementing multiple sensors distributed in array configuration. In this kind of system, some of the sensor types can be identified as photodiodes [1], capacitive sensors [19], radio frequency (RF) sensors [20] [21] and magnetic sensors [22] [23].

Sensor array-based systems can be convenient solutions for the smart surface thanks to their easy integration [1] [23]. Among the existing sensor array systems, the permanent magnets (PM) based measurement solution is often adapted because of its miniaturization feature, its ability to operate without the need of external power source and to measure its

This work was supported by a public grant overseen by the French National Research Agency (ANR) as part of the Parallel Microrobot localized in a large workspace – micro-Spider Program under Grant ANR-19-CE33-0002. (Corresponding author: Gabriel Géron).

Gabriel Géron, Muneeb Ullah Khan, Christine Prelle and Hani Al Hajjar are with Université de technologie de Compiègne, Roberval, Compiègne, France.

Jérémy Terrien is with Université de technologie de Compiègne, service électronique, Compiègne, France.

E-mail: gabriel.geron@utc.fr; jeremy.terrien@utc.fr; muneeb-ullah.khan@utc.fr; hani.al-hajjar@utc.fr; christine.prelle@utc.fr
magnetic field separated by non-magnetic obstacles. Thus, it is
possible to determine the position of the PM, moving in a large
workspace, by interpreting the magnetic fields, measured by the
magnetic sensors array system. The transition from magnetic
measurements to PM position is allowed thanks to different
approaches using numerical solving methods.

The Levenberg-Marquardt algorithm, estimating the real
position of the magnet from all the sensor measurements of the
array by minimizing the mean-square error between the
theoretical and measured magnetic fields, is the most used
approach in this context [24]-[26]. However, the obtained
results reveal localization errors in the millimeter range,
because of the impact of the distribution and the measurement
noise of each sensor on the minimization principle of the mean-
square error. Other methods lead to this same performance, like
Guitron et al. [27], who developed a closed-loop position
feedback control of a miniature robot using 33 Hall effect
sensors array surface, covering an area of 60 × 75 mm². They
used a trilateration method to determine the 2D position of a
PM from the magnetic dipole model, with an average error
equal to 0.995 ± 0.520 mm. Nevertheless, it exists some
exceptions able to reach sub-millimetric localization errors.
Cichon et al. [28] proposed an improved method to determine
the position in the six degrees of freedom of a PM with 16 × 16
3D Hall effect sensor array, covering a 12 × 12 mm² surface.
The method is an inverse magnetostatic problem using an
unscented Kalman filter and an analytical model of the
magnetic source's field to deduce the three positions and the
three orientations of the magnet. The proposed optimal
configuration of the Hall effect sensor array allows them to
obtain an absolute position accuracy equal to 71 µm. In the
same order of magnitude, a previous work [29] has achieved
localization errors of less than 50 µm by using an improved
trilateration method, applicable for a 3D Hall effect sensor array
system.

This paper presents an improved localization method based
on 3D Hall effect sensor array in order to determine the 2D
position of a PM for future micro robotic applications using
smart surfaces. The Newton-Raphson (NR) method is adopted
in addition to mathematical models of the 3-axis magnetic fields
to estimate the 2D position of the PM. These mathematical
models are based on original fine analytical models, describing
accurately the actual magnetic field provided by the PM. These
proposed analytical models take into consideration
experimental uncertainties (orientation of the PM and the
sensors, position shift of the sensing part, performance of the
sensors), determined via a dedicated calibration process. The
combination of these analytical models with the NR method
allows to reduce the influence of the experimental defects and
to accurately estimate the 2D position of the PM.

The paper is organized as follows. Section II introduces the
working principle of the Hall effect sensor array system and the
NR method for the 2D magnetic localization. Section III
describes the experimental setup and the calibration method.
Section IV shows the simulation/experimental results with the
NR method, combined with the calibration method, and a

discussion about the obtained results. Finally, the conclusion is
presented in Section V.

## II. 2D MAGNETIC LOCALIZATION METHOD

Before describing the 2D magnetic localization method, it is
important to first describe the assumptions and the choice made
for the Hall effect sensor array system (e.g. PM type, sensor
type, array distribution, working area, etc.). This is
complemented by a modelling of the magnetic field of the PM
to be localized in order to understand the evolution of the
magnetic field relative to the position of the PM and the sensor.
These preliminary steps are needed for the understanding and
the study of the localization algorithm.

### A. Hall Effect Sensor Array System Overview

The localization principle using a Hall effect sensor array is
based on the use of magnetic field measurements of each sensor
to get the 2D position of a magnetic object (e.g. PM). In the
context of smart surface, such a device has a real interest when
accurate positioning of a magnetic object needs to be achieved.
This interest is enhanced by the miniaturization of the Hall
effect sensors, which facilitates the integration of the array near
the actuation surface, and by the good knowledge of the
magnetic phenomena from a PM. The objective of this paper is
to demonstrate that the new developed localization method
delivers high resolution measurement, useful for micro-robotic
applications, during micro-manipulation or positioning tasks.

To simplify the performance analysis of the localization
system, the study was restricted to the 2D magnetic localization
of only one PM disposed at several positions above a Hall effect
sensor array system (see Fig. 1). Moreover, we supposed that
the faces of the PM are correctly aligned with the reference
frame and that the PM keeps this alignment when it moves from
one position to another one. In this study, the area has been
restricted to an elementary mesh of the Hall effect sensor array,
since it is assumed that the distribution of the localization error
remains the same for all the elementary meshes of the array
when the sensors behave similarly.

The achievement of a high-performance sensor array
requires the use of 3D Hall effect sensors, measuring the
magnetic field of the PM along x-, y- and z-axis. Compared to
the single axis Hall effect sensors, these three measurement

<figure><img src="image_4.png" /><figcaption>**Fig. 1** Application example of the 2D localization of one PM moving in-plane above a Hall effect sensor array system</figcaption></figure>
components of the magnetic field provide additional information about the PM position. Moreover, their distribution into triangular meshes ensures an optimal coverage of the measuring area with the lowest sensor density. Finally, a 5 × 5 × 5 mm³ neodymium cubic PM, with an N50 magnetization, has been chosen, taking benefits of analytical expressions modelling the magnetic field of the cubic PM, useful for the operation of the 2D magnetic localization.

## B. Magnetic Induction Modelling of a Cubic PM

By reducing the PM to a “magnetic charge” distribution, an analytical model of its magnetic flux density can be obtained by solving a Poisson equation, including the magnetostatic field equations [30]. Moreover, we assume that the PM is made of isotropic material and has a magnetization $\vec{M}$ along the vertical component $\vec{z}$ ($\vec{M} = M\vec{z}$). So, a PM with length L, width l, height h and a magnetic induction $\vec{B} = B\vec{z}$ (see Fig. 2), the magnetic inductions of the PM along x-, y- and z-axis $B_x, B_y$ and $B_z$ can be expressed as function of the PM position $p_M = (x_M, y_M, z_M)$ and the sensor’s sensitive point position $p_S = (x_S, y_S, z_S)$ [28] [29]. Thus, these analytical models link the magnetic induction measurements, provided by the 3D Hall effect sensor, to the 2D-position of the PM to be determined.

However, these analytical models are only applied in the ideal scenario where PM and sensors are perfectly aligned and positioned, and for similar performances of the sensors. In real scenario, the magnetic measurements depend on the uncertainty sources linked to the positioning and the operation of PM and sensors.

## C. Consideration of the Experimental Uncertainties

Despite previous assumptions, in real scenario, experimental defects alter the magnetic field profile of the field, measured by the sensors, resulting in misfit between theoretical models and real quantities. To solve this issue, we need to build an improved model of the magnetic measurements, considering experimental uncertainty sources, in order to reduce the impact of these experimental defects. Building such a model requires to know the types of uncertainty, impacting on magnetic measurements, and their associated analytical expressions. In this case, three types of uncertainties are mainly identified.

The first uncertainty type is related to the alignment of the cubic PM relatively to the reference frame (see Fig. 3), linked to the sensor’s PCB plane. Even if the PM is wisely placed in

<figure><img src="image_5.png" /><figcaption>Fig. 2 PM with the magnetization along z-axis and sensor sensing point in the reference frame (O, x, y, z)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3 PM oriented along x-, y- and z-axis and sensor sensing point in the reference frame (O, x, y, z)</figcaption></figure>

order to align the magnet frame with the reference frame, there may exist slight orientation errors of the magnet. By considering these slight orientations, defined by the approximated rotation matrix $R_{\theta_M}$ (see Appendix), the analytical models are modified as:

$$ \begin{pmatrix} B_{ox} \\ B_{oy} \\ B_{oz} \end{pmatrix} = R_{\theta_M} \begin{pmatrix} B_x (R_{\theta_M}^T p_M^T) \\ B_y (R_{\theta_M}^T p_M^T) \\ B_z (R_{\theta_M}^T p_M^T) \end{pmatrix} \quad (1) $$

The second type refers to the black box effect of the sensor, preventing to know the precise disposition of each sensing part inside the 3D Hall effect sensor. Although the datasheet of the sensor [31] provides information about this, the position and the orientation of each sensing part are not the same and change from one sensor to another. In this case, for one sensor, we considered the position shifts $\Delta p$ and the approximated matrix rotation $R_{\theta_S}$ (see Appendix) for each sensing part, modifying the analytical models as:

$$ \begin{pmatrix} B_{bbx} \\ B_{bby} \\ B_{bbz} \end{pmatrix} = R_{\theta_S} \begin{pmatrix} B_x(p_M + \Delta p) \\ B_y(p_M + \Delta p) \\ B_z(p_M + \Delta p) \end{pmatrix} \quad (2) $$

Finally, the third and last type concerns the specific features of each sensor. In this context, the measurement noise is not considered, even if it has an impact on the magnetic measurement resolution and therefore on the estimation performance of the 2D position of the PM [29]. Indeed, it can be reduced by averaging a number of measurements to enhance the signal to noise ratio. However, the characteristic of the magnetic measurement may differ between the sensors because of several factors intrinsic to the sensor (manufacturing tolerance, semiconductor imperfections, aging, wear, etc.). Particularly, the measuring sensibility and the offset relative to the real magnetic field may change from one sensor to another. In this case, it can become difficult to extract information from multiple sensors. Moreover, the relationship between the real magnetic field and the sensor output measurement is not exactly linear, due to the Hall effect principle. All these specific features lead to a polynomial regression between the actual and measured magnetic inductions. In our case, we have chosen to use a 4th order polynomial regression, including the polynomial coefficients $(a_0, a_1, a_2, a_3, a_4)$, which induces new analytical models of the magnetic measurements as follows:
$$ \begin{pmatrix} B_{nl_x} \\ B_{nl_y} \\ B_{nl_z} \end{pmatrix} = \begin{pmatrix} a_4 B_x^4 + a_3 B_x^3 + a_2 B_x^2 + a_1 B_x + a_0 \\ a_4 B_y^4 + a_3 B_y^3 + a_2 B_y^2 + a_1 B_y + a_0 \\ a_4 B_z^4 + a_3 B_z^3 + a_2 B_z^2 + a_1 B_z + a_0 \end{pmatrix} \quad (3) $$

The combination of the equations (1), (2) and (3) gives a complete analytical model of the experimental magnetic measurements, provided by the output of the sensor, as:

$$ \begin{pmatrix} B_{S_x} \\ B_{S_y} \\ B_{S_z} \end{pmatrix} = \begin{pmatrix} a_4 \tilde{B}_{S_x}^4 + a_3 \tilde{B}_{S_x}^3 + a_2 \tilde{B}_{S_x}^2 + a_1 \tilde{B}_{S_x} + a_0 \\ a_4 \tilde{B}_{S_x}^4 + a_3 \tilde{B}_{S_x}^3 + a_2 \tilde{B}_{S_x}^2 + a_1 \tilde{B}_{S_x} + a_0 \\ a_4 \tilde{B}_{S_x}^4 + a_3 \tilde{B}_{S_x}^3 + a_2 \tilde{B}_{S_x}^2 + a_1 \tilde{B}_{S_x} + a_0 \end{pmatrix} \quad (4) $$

With:

$$ \begin{pmatrix} \tilde{B}_{S_x} \\ \tilde{B}_{S_y} \\ \tilde{B}_{S_z} \end{pmatrix} = R_{\theta_S} R_{\theta_M} \begin{pmatrix} B_x (R_{\theta_M}^T (p_M + \Delta p)^T) \\ B_y (R_{\theta_M}^T (p_M + \Delta p)^T) \\ B_z (R_{\theta_M}^T (p_M + \Delta p)^T) \end{pmatrix} \quad (5) $$

Consequently, the objective of the mathematical expressions (4) and (5) is to build an exact model of the magnetic induction measurements at sensor output. This upgrades the relationship between the magnetic measurements, faced with uncertainty sources, and the 2D position of the PM to be estimated. Thus, using these actual models in an appropriate calculation algorithm will reduce the localization errors and will improve the estimation of the 2D position of the PM.

### D. Newton-Raphson Method for 2D Localization

Numerical methods are more appropriate for the estimation of the 2D-position of the PM. By analyzing the magnetic models (4) and (5), it is very difficult to find unique mathematical function expressing the 2D position of the PM from the magnetic induction measurements. In this context, the Newton-Raphson (NR) method has been chosen, among other numerical methods, thanks to its ability to handle non-linear equations, the small number of parameters to be set and its fast convergence speed. The NR method consist in an iterative algorithm, which allows to estimate unknown values $v_1, ..., v_n$ by minimizing a non-linear function $F$, i.e. find $v_1, ..., v_n$ such that $F(v_1, ..., v_n) \approx 0$.

In our case, the function $F$ is the error between the analytical magnetic induction models (4) and (5) in the sensor-base and the magnetic measurements provided by three sensors of a mesh for a known elevation $z_M$:

$$ F(x_M, y_M) = \begin{pmatrix} B_{S_x}^i - B_{m_x}^i \\ B_{S_y}^i - B_{m_y}^i \\ B_{S_z}^i - B_{m_z}^i \end{pmatrix}_{i=1,2,3} \in \mathbb{R}^9 \quad (6) $$

Where $B_{m_x}^i, B_{m_y}^i$ and $B_{m_z}^i$ are the magnetic measurements for the $i^{th}$ sensor along x-, y- and z-axis, respectively.

Assuming that the application $F$ is differentiable and has at least one root, the solution of the equation $F(x_M, y_M) = 0$ is iteratively calculated as follow:

$$ \left( \frac{x_M^{k+1}}{y_M^{k+1}} \right) = \left( \frac{x_M^k}{y_M^k} \right) - J^+ F(\tilde{x}_M^k, \tilde{y}_M^k) \quad (7) $$

With $(\tilde{x}_M^k, \tilde{y}_M^k)$ the estimated 2D position of the PM after $k$ iterations and $J$ the Jacobian matrix of $F$:

$$ J = \frac{\frac{\partial B_{S_x}^i}{\partial x_M} \frac{\partial B_{S_x}^i}{\partial y_M}}{\frac{\partial B_{S_y}^i}{\partial x_M} \frac{\partial B_{S_y}^i}{\partial y_M}} \left[ \frac{\partial B_{S_z}^i}{\partial x_M} \frac{\partial B_{S_z}^i}{\partial y_M} \right]_{i=1,2,3} \in \mathbb{R}^{9 \times 2} \quad (8) $$

$J^+$ corresponds to the Moore-Penrose inverse of the Jacobian matrix $J$, which is used instead of the inverse of this matrix because, in our case, $J$ is not a square matrix: $J^+ = (J^T J)^{-1} J^T$. The calculation of the 2D position of the PM at the $k^{th}$ iteration is carried out from an initial position $(\tilde{x}_M^0, \tilde{y}_M^0)$, equal to the position of the barycenter of the triangular mesh in order to ensure a precise and fast convergence of the estimation of the 2D position of the PM:

$$ \left\{ \begin{aligned} \tilde{x}_M^0 &= \frac{x_{S_1} + x_{S_2} + x_{S_3}}{3} \\ \tilde{y}_M^0 &= \frac{y_{S_1} + y_{S_2} + y_{S_3}}{3} \end{aligned} \right. \quad (9) $$

This method could represent an issue when singularities exist in the determination of the inverse matrix $J^T J$ used to calculate $J^+$. These singularities occur when the determinant of the matrix $J^T J$ is equal to 0 for a given position $(x_M, y_M)$ and makes it impossible to estimate the 2D position of the PM. Consequently, the operation of the NR method proceeds properly for 2D positions of the PM with $\det(J^T J) \neq 0$, which induces the following condition:

$$ \sum_{k=1}^{8} \sum_{l=k+1}^{9} (J_{kl} J_{l1} - J_{k2} J_{l1})^2 \neq 0 \quad (10) $$

Where $(J_{ij})_{\substack{1 \le i \le 9 \\ 1 \le j \le 2}}$ are the values of the Jacobian matrix $J$ in the $i^{th}$ row and in the $j^{th}$ column. The respect of the condition (10) depends on two parameters linked to the geometrical disposition of the 3D Hall effect sensor array and the PM (see Fig. 4): the size of the triangular elementary mesh ($d$) and the elevation of the PM relative to the sensors ($z$).

So, the 2D magnetic localization consists in applying the NR method from the magnetic induction measurements provided by the three sensors of the identified elementary triangular mesh, where the PM is located on the 3D Hall effect sensor array (see

<figure><img src="image_7.png" /><figcaption>Fig. 4 Possible $d$ and $z$ values for which the condition (10) is satisfied (green zone) or not (red zone)</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5** Localization principle of the PM inside an elementary mesh by estimating its 2D-position with the NR method</figcaption></figure>

Fig. 5). This identification process selects the elementary mesh recording the strongest magnetic field from the PM. Thus, the NR algorithm estimates the final values of the 2D-position of the PM for 100 iterations of the recursive process ($\overline{(x_M^{100}, y_M^{100})}$). We have chosen these final values in order to ensure a precise and homogeneous convergence of the recursive process at each fixed of the cubic PM. Moreover, as the calculation time at each iteration *k* is constant in the NR algorithm, this number of 100 iterations remains to keep a fast convergence of the recursive process.

### III. EXPERIMENTAL DESIGN

#### A. Experimental Setup

The experimental setup consists of a Printed Circuit Board (PCB) with nine 3D digital Hall effect sensors (TMAG5170-A2 from Texas Instruments) (see Fig. 6(a)), controlled by a microcontroller (LPC804 from NXP) based on Arm-Cortex-M0+. This microcontroller establishes a Serial Peripheral Interface (SPI) communication to collect the required magnetic measurement from input parameters (sensor designation, axis, etc.) imposed by user. The output of each sensor provides a 12-bit value, including the three components of the magnetic field with a magnetic range between -300 mT and 300 mT. The selected sensor delivers a measurement of the magnetic field components with 1-sigma magnetic noise equal to 28 µT for the x- and y-component and 13 µT for the z-component. The distribution of the nine sensors on the PCB was carried out so as to make three elementary equilateral triangular meshes (6 mm, 16 mm and 26 mm in size), in order to assess the performance of the magnetic localization method through several configurations of the Hall effect sensor array.

A Labview program has been developed to collect the digital values of the three components of the magnetic field of the PM, converted to decimal values, with an acquisition time equal to 16 ms per sensor (see Fig. 6(b)). The measurements are then used in the NR algorithm to calculate the 2D relative position of the PM in an elementary mesh with a total calculation time of 15 ms for 100 iterations of the recursive process. The relative position of the PM is defined with respect to the position of the

<figure><img src="image_9.png" /><figcaption>**Fig. 6** (a) PCB with nine TMAG5170-A2 digital Hall effect sensors distributed as three triangular elementary meshes (6 mm, 16 mm and 26 mm in size), (b) Working principle of the measurement of the PM magnetic field with one 3D Hall effect sensor</figcaption></figure>

leftmost sensor in the elementary triangular mesh, corresponding to the reference position (0,0).

A three axes positioning system was implemented that consists of two motorized stages and one manual stage. It is used to position the PM above the sensor PCB (see Fig. 7(a)). The two motorized stages are linear stages (UMR8.25 from Newport) with motorized actuator (TRA25CC from Newport) to deliver high precision x-/y-axis stroke up to 25 mm. A motion controller is used to drive each motorized actuator (see Fig. 7(b)) that can also sense the stroke measured with the help of a built-in encoder in the actuator, with a resolution equal to 30 nm. These measurements are used as references to establish the estimation errors of the 2D position of the PM, determined by the NR method. The manual stage is a linear stage (UMR5.16 from Newport) with a micrometer head (BM11.16 from Newport), used to position the PM at different altitude *z* from the sensor PCB. This manual stage can perform up to 16 mm-stroke with a positioning resolution equal to 20 µm. The cubic PM is precisely stuck on a plexiglass plate, which is fixed with the help of three axes positioning system over the sensor PCB as shown in Fig. 7(a).

#### B. Calibration Process

From the analytical models of the magnetic induction considering the experimental uncertainties, it remains to determine all the parameters associated to these experimental uncertainties from a calibration process. For that, each sensor is calibrated from a single measuring area, also used to quantify the efficiency of the 2D magnetic localization in the three elementary meshes later. In this area, 224 (16 × 14) evenly distributed measurement points, spaced by 1 mm, have been chosen to calibrate the sensors (see Fig. 8). The choice of this distribution of the measurement points relies on the space between two successive measurement points, which has an
<figure><img src="image_10.png" /><figcaption>Fig. 7 (a) Experimental setup to assess the performance of the 2D magnetic localization of a cubic NdFeB PM with the NR algorithm, (b) Working principle of the control of the three axes positioning system to carry out the 2D positioning of the cubic PM at elevation z</figcaption></figure>

influence on the efficiency of the calibration process. Thus, the closer the sensors are, the more accurately the calibration will determine the uncertainty parameters. Nevertheless, to optimize the efficiency of the calibration process and the collection time of the data measurements, a space of 1 mm between two measurement points has been chosen. Because of the stroke limitation of the motorized actuators included in the 3-axis positioning system, the chosen calibration workspace covers only the two smallest triangular elementary meshes with a rectangular shape scan in order to facilitate the displacement from one measuring point to another. Each measuring point delivers the x-, y- and z-magnetic field intensities of each sensor and they are assigned in data matrices $B_{cx}^i$, $B_{cy}^i$, $B_{cz}^i$, corresponding to all the magnetic field measurements for the $i^{th}$ sensor.

From the obtained measurement data, Levenberg-Marquardt algorithm has been chosen to estimate the parameters associated to the experimental uncertainties. This numerical optimization algorithm is adapted to adjust the data measurements with the theoretical models by solving non-linear least square problems with a large number of uncertainties parameters (a total of 40). The objective of this algorithm is to find iteratively a numerical solution of these parameters in order to minimize the square error between the theoretical models of the magnetic field and the obtained measurements for each sensor. In other words, for the $i^{th}$ sensor, the algorithm estimates an optimal parameter list $\alpha^*$ such as:

<figure><img src="image_11.png" /><figcaption>Fig. 8 Calibration principle of the six sensors in a single workspace including 224 measurement points, each delivering the three components of the magnetic induction of each sensor</figcaption></figure>

$$ \alpha^* = \operatorname*{argmin}_{\left( \sum_{p=1}^{N} \left\| \begin{pmatrix} B_{S_x}^i - B_{c_x}^i \\ B_{S_y}^i - B_{c_y}^i \\ B_{S_z}^i - B_{c_z}^i \end{pmatrix} \right\|_2^2 \right)} \quad (11) $$

For each sensor, these estimated parameters are then injected in the assigned analytical models to carry out in the best conditions the 2D positioning of the PM. It remains to be verified whether the calibration method has a real impact on the performance of the 2D magnetic localization system.

## IV. RESULTS

In this part, four configurations have been chosen to evaluate the performances of the 2D magnetic localization: (z = 4 mm, d = 6 mm), (z = 4 mm, d = 16 mm), (z = 10 mm, d = 6 mm) and (z = 10 mm, d = 16 mm). For these four configurations, the performance assessment is determined by simulation and then experimentally.

### A. Performance Analysis of 2D Localization Estimation Using Newton-Raphson Method

Initially, for the simulated calculation, we considered that the 2D magnetic localization with NR algorithm is applied in the ideal conditions where the experimental uncertainties are ignored (i.e., perfect alignment and positioning between PM and sensors, no non-linearity error, etc.). The interest of this consideration is to study the influence of the experimental uncertainties on the performances of the 2D magnetic localization thereafter, by comparing the results obtained in ideal conditions with those obtained by considering the different uncertainty sources in the simulated calculation.

The x- and y-position values of the PM were calculated with a Matlab program, running the NR algorithm inside an elementary mesh. The measurements of each sensor were included in this program by incorporating the analytical models of the magnetic field and a white noise corresponding to the measurement noise of the TMAG5170-A2 Hall effect sensors, defined in the datasheet:
$$ \overrightarrow{B_{simu}} = \begin{pmatrix} B_{S_x} + \sigma_{S_x} \\ B_{S_y} + \sigma_{S_y} \\ B_{S_z} + \sigma_{S_z} \end{pmatrix} \text{ with } \begin{cases} \sigma_{S_x}, \sigma_{S_y} \sim N(0; 28 \mu T) \\ \sigma_{S_z} \sim N(0; 13 \mu T) \end{cases} \quad (12) $$

In order to evaluate the performances of the NR algorithm on the 2D magnetic localization, we determined the estimation error of the x- and y-position of the PM for each position $P(x_p, y_p)$ inside the elementary mesh. Here, the Mean Absolute Error (MAE) has been chosen to define the localization error along x- and y-axis at each position $P(x_p, y_p)$:

$$ MAE = \frac{\|x_{Pcal} - x_{Preal}\| + \|y_{Pcal} - y_{Preal}\|}{2} \quad (13) $$

Where $(x_{Pcal}, y_{Pcal})$ is the 2D estimated position of the PM and $(x_{Preal}, y_{Preal})$ is the 2D theoretical position of the PM at the position $P(x_p, y_p)$.

Fig. 9 shows color maps of the MAE, established by simulation, inside an elementary mesh in the four pre-defined configurations with three successive averaged measurements for each sensor in ideal conditions. This number of measurements has been chosen to impose a great signal to noise ratio while keeping a suitable acquisition time for the localization (about 160 ms for the reading of the three sensors and the computation of the 2D position of the PM). Given the non-homogeneous nature of the MAE distribution inside the elementary mesh, the maximum MAE is considered as indicator about the precision of the proposed 2D magnetic localization.

For each configuration, this maximum MAE was calculated with 100 successive color maps of the MAE for each configuration. So, the average and the standard deviation of the 100 calculated maximum MAE is listed as function of the configuration of the sensor array system, as shown in TABLE I in the simulation results (ideal scenario). The results clearly show a correlation between the distance PM/sensors and the precision of the 2D magnetic localization, with localization errors remaining very low relative to the PM size.

## B. Experimental Results

2D magnetic localization using the NR method is applied experimentally without and with correction of the experimental uncertainties. Before calculating the 2D position of the PM, the calibration parameters were determined for each sensor on the PCB by using the calibration method previously detailed in section III. These calibration parameters can be then injected in the NR algorithm, when the experimental uncertainties are taken into consideration for the calculation of the 2D position of the PM. The interest of this calibration process is obviously to improve the effectiveness of the localization system by replicating the real magnetic measurement. Moreover, simulated results show that the addition of these analytical models (real scenario), completed with the calibration parameters, has weak impact on the performance of the 2D magnetic localization system (TABLE I).

<figure><img src="image_12.png" /><figcaption>Fig. 9 Color maps of the simulated and experimental MAE (in µm) of the 2D-position of the PM inside the triangular elementary mesh for four different configurations, depending on the mesh size (d) and the PM elevation (z)</figcaption></figure>
From the experimental setup, the MAE of the 2D position of the PM is experimentally determined by comparing the 2D positions estimated by the NR algorithm with the reference 2D positions from the encoders of the motorized actuators (detail in Section III.A.), inside the triangular elementary mesh. The maximum MAE from the experimental setup is determined and listed in TABLE I for each configuration (experimental results), when the experimental uncertainties are considered with calibration or without calibration in the NR algorithm. As shown in TABLE I, this clearly demonstrates that the calibration process greatly improves the performance of the 2D magnetic localization with the NR method. Moreover, the experimental results match well with the simulated results when the uncertainties are incorporated in the calculation. However, this match is less effective for the first two configurations (n°1 and n°2) in the case where the PM is close to the sensors, with experimental errors well above the theoretical performances. The non-consideration of other experimental uncertainties (non-perfect alignment of the motorized actuators, external magnetic fields, ambient temperature, etc.) could explain this discrepancy in results and is stronger when large magnetic field are measured.

Finally, Fig. 9 shows color maps of the experimental MAE of each 2D position of the PM inside an elementary mesh in the same four pre-defined configurations with three successive averaged measurements for each calibrated sensor. Compared to the simulation color maps, the non-homogeneous nature of the localization error inside the mesh is again present, even if the maximum localization errors are not located at the same spot.

<table><caption>TABLE I
MAXIMUM THEORETICAL AND EXPERIMENTAL MAE OF THE 2D POSITION OF THE PM WITH THE CONSIDERATION OF THE EXPERIMENTAL UNCERTAINTIES</caption><thead><tr><th rowspan="2">CONFIGURATION</th><th colspan="2">SIMULATION RESULTS (µm)</th><th colspan="2">EXPERIMENT RESULTS (µm)</th></tr><tr><th>Ideal scenario</th><th>Real scenario</th><th>No calibration</th><th>With calibration</th></tr></thead><tbody><tr><td>N°1</td><td>1.281 ± 0.480</td><td>1.319 ± 0.456</td><td>122.689</td><td>4.327</td></tr><tr><td>N°2</td><td>7.275 ± 2.259</td><td>7.860 ± 2.793</td><td>149.038</td><td>10.121</td></tr><tr><td>N°3</td><td>12.066 ± 3.564</td><td>13.548 ± 4.518</td><td>176.682</td><td>12.618</td></tr><tr><td>N°4</td><td>25.367 ± 6.459</td><td>26.849 ± 7.575</td><td>268.415</td><td>23.249</td></tr></tbody></table>

Finally, an experimental test of the cubic PM localization was carried out to check the performance of the 2D magnetic localization, when the PM is placed at different position above a 3D Hall effect sensor array. The prototype includes a PCB of 23 TMAG5170-A2 sensors arranged in an array distribution, controlled by the same microcontroller and using the same acquisition process than the previous PCB of 9 TMAG5170-A2 (see Fig. 10(a)). As previously, the elementary meshes of the 3D Hall effect sensor array are triangular and the distance between two successive sensors is equal to 6 mm. The 3-axis positioning system is used to place the PM above the 3D Hall effect sensor at different position, describing a “snail” trajectory

<figure><img src="image_13.png" /><figcaption>Fig. 10 (a) Setup of the calibration and test phases for the 2D localization of a PM above a prototype of 3D Hall effect sensor array, (b) MAE of the each estimated 2D-position of the PM in the case where the sensors are calibrated or not</figcaption></figure>

on a 24 × 21 mm² measuring area, with an altitude of 10 mm relative to the sensor array. Before the experimental test, a calibration process was carried out with the same number of measurement points (224) to keep the same collection time of the data measurements, imposing a space of 1.5 mm between two successive measurement points. The 2D position of the PM is estimated at each point of the trajectory with a gap of 3 mm between each point. These estimated positions are compared with the reference 2D position measured by the encoders of the motorized actuators. The MAE are presented in Fig. 10(b) in the case where the sensors have been or not previously calibrated with the aim of showing the influence of the consideration of the experimental uncertainties on the precision of the 2D magnetic localization.

### C. Discussion

Based on initial simulation results, the NR method can ensure a high precision of the 2D magnetic localization relative to the selected magnet inside an elementary mesh. Indeed, the obtained maximum localization errors can range from a few micrometers to a few tens of micrometer as function of the distances *d* and *z*. Assuming that these localization errors are reproducible for a Hall effect sensor array, it would therefore be possible to design a magnetic localization system with a relatively low Hall effect sensor density, distributed in an array, while keeping a high precision on the estimation of the 2D position of the magnet. Obviously, it is allowed, as long as the distances *d* and *z* fulfill the condition (10) to avoid the singularities of the pseudo-inverse of the Jacobian matrix in the NR algorithm.

The experimental results, presented in TABLE I, show how the consideration of the experimental uncertainties is essential to achieve a 2D high-precision magnetic localization. Moreover, the similar simulated and experimental results with
the use of calibration prove that the magnetic field models, integrating the experimental uncertainties, nearly define the real magnetic field induced by the PM. In this case, most experimental uncertainties have been well identified and well modeled, but the gap between the simulated and experimental results for the first configuration points out that it remains other parameters to clarify (e.g., temperature variation, alignment and repeatability of each actuator of the 3-axis positioning system, etc.). Nevertheless, through these results, it is important to take in consideration the calibration of the sensors in the 2D magnetic localization method for applications requiring an open- and closed-loop control with high precision.

Finally, the experimental test results, presented in Fig. 10, show that the performances of the 2D magnetic localization remain the same on the entire 3D Hall effect sensor array. Moreover, it also shows how taking into account the experimental uncertainties in the NR algorithm reduces drastically the estimation errors on the 2D position of the magnet. Without the calibration of the sensors, the estimated MAE of the 2D position of the magnet is in average 127.50 µm, whereas it is in average 12.73 µm with the calibration of the sensors, a decrease of 90 %. To enhance the effectiveness of the used method, other methods using Hall effect sensor array are listed in TABLE II in order to compare their real performances with the proposed method. We have specifically chosen examples of Hall effect sensor system, which localize a PM, including measurement noise and the sensor density, to properly compare the real localization errors. In these conditions, our 2D magnetic localization method provides better performance with a great decrease of the localization errors, by using only three sensors for the calculation of the 2D position of the PM. Moreover, even if the measurement noise always influences the localization error, the combination of the NR method with the fine analytical models improves considerably the precision of the position estimation of the PM. Thanks to this, the impact of the measurement noise is reduced and the localization errors tend towards the performance in the ideal scenario.

<table><caption>TABLE II
EXPERIMENTAL TEST PERFORMANCE OF THE 2D MAGNETIC LOCALIZATION WITH HALL EFFECT SENSOR ARRAY BY USING OUR METHOD COMPARED TO OTHER PUBLISHED WORKS</caption><thead><tr><th>2D magnetic localization method</th><th>Measurement noise (µT)</th><th>Number of sensors/mm²</th><th>Localization error (µm)</th></tr></thead><tbody><tr><td>Levenberg-Marquardt [24]</td><td>90</td><td>0.02</td><td>2020</td></tr><tr><td>Unscented Kalman filter [28]</td><td>40</td><td>0.1</td><td>71</td></tr><tr><td>Classical trilateration [27]</td><td>/</td><td>0.01</td><td>995</td></tr><tr><td>Improved trilateration [29]</td><td>7.5</td><td>0.08</td><td>45.7</td></tr><tr><td>Newton-Raphson (proposed method)</td><td>13</td><td>0.08</td><td>12.73</td></tr></tbody></table>

V. CONCLUSION

In this paper, a new localization method has been presented using a 3D Hall effect sensor array system in order to localize a

PM. This method applies the Newton Raphson algorithm to determine iteratively the 2D position of this PM inside an elementary mesh of the sensor array with high precision and with a very fast calculation speed. The algorithm is complemented by new fine mathematical models of the magnetic induction provided by the PM, including uncertainty parameters to reduce the influence of the experimental defects in real scenario. The theoretical and experimental results show good performances of the 2D magnetic localization method with localization errors ranging from a few micrometers to a few tens of micrometer. This demonstrates that the consideration of the measurement uncertainties, determined by an adapted calibration process of the sensors, allows to estimate with high precision the 2D position of the PM. Finally, with this localization method, we are also able to localize the PM, placed at different position of a specific trajectory, above a prototype of a 3D Hall effect sensor array with an average precision of 12.73 µm on a 24 × 21 mm² measuring area.

In perspective, we will plan to develop a 2D tracking method, which allows to control in real time the 2D position of the PM moving above the 3D Hall effect sensor system. The resulting tracking process has to be optimally designed to estimate the 2D position of the PM with high precision and speed, since the measured frequency will limit the dynamics of the PM. Thus, by combining the localization and tracking process, the 3D Hall effect sensor array system could be set up for closed-loop control for smart surface applications. Moreover, we could modify the NR method to improve the 2D magnetic localization by estimating the orientation and the altitude of the PM. In this case, if the PM changes its orientation and its altitude, we can update the analytical models by considering these modifications and improve the efficiency of the 2D magnetic localization.

APPENDIX A

First-order approximation of the rotation matrix $R_θ$:

By assuming that an object carries out rotations in space along z-axis ($\theta_z$) first, then y-axis ($\theta_y$) and finally x-axis ($\theta_x$) (see Fig. 3), the 3 × 3 matrix rotation $R_\theta$, associated with this object, is defined by:

$$R_{\theta} = R_{\theta_x} R_{\theta_y} R_{\theta_x}$$

Where $R_{\theta_x}$, $R_{\theta_y}$ and $R_{\theta_z}$ are the 3 × 3 rotation matrices, associated with the object, along x-axis, y-axis and z-axis, respectively. These rotation matrices are defined by:

$$R_{\theta_x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & c\theta_x & -s\theta_x \\ 0 & s\theta_x & c\theta_x \end{bmatrix} \\ R_{\theta_y} = \begin{bmatrix} c\theta_y & 0 & s\theta_y \\ 0 & 1 & 0 \\ -s\theta_y & 0 & c\theta_y \end{bmatrix} \\ R_{\theta_z} = \begin{bmatrix} c\theta_z & -s\theta_z & 0 \\ s\theta_z & c\theta_z & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Where $\theta_x$, $\theta_y$ and $\theta_z$ represent the rotation angles of the object along x-axis, y-axis and z-axis, also called yaw, pitch and roll angles, respectively. $c\theta$ and $s\theta$ are the abbreviations of the cosine and sine function at the $\theta$ value: $c\theta = \cos \theta$ and $s\theta = \sin \theta$.
By assuming that $\theta_x$, $\theta_y$ and $\theta_z$ are very small and knowing that $c\theta \approx 1$ and $s\theta \approx \theta$ when $\theta \ll 1$, the first-order approximation of the rotation matrix $R_\theta$ is equal to:

$$R_{\theta} = \begin{bmatrix} 1 & -\theta_{z} & \theta_{y} \\ \theta_{z} & 1 & -\theta_{x} \\ -\theta_{y} & \theta_{x} & 1 \end{bmatrix}$$

## ACKNOWLEDGMENT

This work was supported by a public grant overseen by the French National Research Agency (ANR) as part of the Parallel Microrobot localized in a large workspace – micro-Spider Program under Grant ANR-19-CE33-0002.

## REFERENCES

[1] M. Ataka, M. Mita, H. Fujita, "Autonomous decentralized operations of stack-integrated sensor/actuator array for 2D planar micro manipulator", 2012 IEEE 25th International Conference on Micro Electro Mechanical Systems (MEMS), Paris, France, 2012, pp. 1133-1136, doi: 10.1109/MEMSYS.2012.6170362

[2] Y. Fukuta, Y.-A. Chapuis, Y. Mita, H. Fujita, "Design, fabrication, and control of MEMS-based actuator arrays for air-flow distributed micromanipulation", Journal of Microelectromechanical Systems, vol. 15, no. 4, pp. 912-926, Aug. 2006, doi: 10.1109/JMEMS.2006.879378

[3] R. Yahiaoui, R. Zeggari, J. Malapert, J.-F. Manceau, "A MEMS-based pneumatic micro-conveyor for planar micromanipulation", Mechatronics, vol. 22, no. 5, pp. 515-521, Aug. 2012, doi: 10.1016/j.mechatronics.2011.04.005

[4] A. Delettre, G. J. Laurent, N. Le Fort-Piat, "A new contactless conveyor system for handling clean and delicate products using induced air flows", 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan, 2010, pp. 2351-2356, doi: 10.1109/IROS.2010.5652194

[5] K. Georgia, A. Nikos, "Micro-Manipulation Methods and Assembly of Hexagonal Microparts on a Programmable Platform with Electrostatic Forces", International Journal of Mechanics and Control, vol. 20, no. 1, pp. 71-80, 2019

[6] T. Hosobata, A. Tamamoto, T. Higuchi, "Transparent Synchronous Electrostatic Actuator for Long-Stroke Planar Motion", IEEE/ASME Transactions on Mechatronics, vol. 20, no. 4, pp. 1765-1776, Sep. 2014, doi: 10.1109/TMECH.2014.2353815

[7] M. Ataka, B. Legrand, L. Buchaillot, D. Collard, H. Fujita, "Design, Fabrication, and Operation of Two-Dimensional Conveyance System With Ciliary Actuator Arrays", IEEE/ASME Transactions on Mechatronics, vol. 14, no. 1, pp. 119-125, Feb. 2009, doi: 10.1109/TMECH.2008.2004770

[8] J. W. Suh, R. B. Darling, K.-F. Bohringer, B. R. Donald, H. Baltes, G. T. A. Kovacs, "CMOS integrated ciliary actuator array as a general-purpose micromanipulation tool for small objects", Journal of Microelectromechanical Systems, vol. 8, no. 4, pp. 483-496, Dec. 1999, doi: 10.1109/84.809064

[9] L. Petit, A. Hassine, J. Terrien, F. Lamarque, C. Prelle, "Development of a Control Module for a Digital Electromagnetic Actuators Array", IEEE Transactions on Industrial Electronics, vol. 61, no. 9, pp. 4788-4796, Nov. 2013, doi: 10.1109/TIE.20132290755

[10] S. Ozasa, S. B. Kim, H. Nakano, M. Sawae, H. Kobayashi, "The Study on the Shape of 2-D Stator with Electromagnets and Permanent Magnets for 3-D Superconducting Actuator", Physics Procedia, vol. 65, pp. 257-260, 2015, doi: 10.1016/j.phpro.2015.05.141

[11] N. Arora, M. U. Khan, L. Petit, F. Lamarque, C. Prelle, "Design and Development of a Planar Electromagnetic Conveyor for the Microfactory", IEEE Transactions on Mechatronics, vol. 24, no. 4, pp. 1723-1731, May. 2019, doi: 10.1109/TMECH.2019.2917007

[12] K. Boutoustous, G. J. Laurent, E. Dedu, L. Matignon, J. Bourgeois, N. Le Fort-Piat, "Distributed control architecture for smart surfaces", 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan, 2010, pp. 2018-2024, doi: 10.1109/IROS.2010.5650668

[13] P. Cheng, C.-H. Menq, "Ultraprecise Three-Axis Visual Motion Tracking of Microscopic Objects", IEEE Transactions on Instrumentation and

Measurement, vol. 66, no. 10, pp. 2597-2605, Jun. 2017, doi: 10.1109/TIM.2017.2712939

[14] A. N. André, P. Sandoz, B. Mauzé, M. Jacquot, G. J. Laurent, "Sensing One Nanometer Over Ten Centimeters: A Microencoded Target for Visual In-Plane Position Measurement", IEEE/ASME Transactions on Mechatronics, vol. 25, no. 3, Jan. 2020, doi: 10.1109/TMECH.2020.2965211

[15] E. Yusifli, R. Yahiaoui, S. M. Qaisar, M. Addouche, B. Al-Mahdawi, H. Bourouina, G. Herlem, T. Gharbi, "FPGA-Based HD Camera System for the Micropositioning of Biomedical Micro-Objects Using a Contactless Micro-Conveyor", Micromachines, vol. 8, no. 3, p. 74, Mar. 2017, doi: 10.3390/mi8030074

[16] G. Su, G. Lu, P. Yan, "Planar Motion Measurement of a Compliant Micro Stage: An Enhanced Microscopic Vision Approach", IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 5, pp. 1930-1939, Jun. 2019, doi: 10.1109/TIM.2019.2924066

[17] Z. Xu, G. Han, H. Du, X. Wang, J. Liu, Y. Yang, "A Generic Algorithm for Position-Orientation Estimation With Microscopic Vision", IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1-10, May. 2022, doi: 10.1109/TIM.2022.3176893

[18] H. Yamamoto, T. Sano, "Study of micromanipulation using stereoscopic microscope", IEEE Transactions on Instrumentation and Measurement, vol. 51, no. 2, pp. 182-187, Apr. 2002, doi: 10.1109/19.997809

[19] A. Romani, N. Manaresi, L. Marzocchi, G. Medoro, A. Leonardi, L. Altomare, M. Tartagni, R. Guerrieri, "Capacitive sensor array for localization of bioparticles in CMOS lab-on-a-chip", 2004 IEEE International Solid-State Circuits Conference (IEEE Cat. No.04CH37519), San Francisco, CA, USA, 2004, pp. 224-225, doi: 10.1109/ISSCC.2004.1332675

[20] J. Hou, Y. Zhu, L. Zhang, Y. Fu, F. Zhao, L. Yang, G. Rong, "Design and Implementation of a High Resolution Localization System for In-Vivo Capsule Endoscopy", 2009 Eighth IEEE International Conference on Dependable, Autonomic and Secure Computing, Chengdu, China, 2009, pp. 209-214, doi: 10.1109/DASC.2009.110

[21] D. A. Duecker, T. Johannink, E. Kreuzer, V. Rausch, E. Solowjow, "An Integrated Approach to Navigation and Control in Micro Underwater Robotics using Radio-Frequency Localization", 2019 International Conference on Robotics and Automation (ICRA), Montreal, Canada, 2019, pp. 6846-6852, doi: 10.1109/ICRA.2019.8794088

[22] C. Hu, M. Li, S. Song, W. Yang, R. Zhang, M. Q.-H. Meng, "A Cubic 3-Axis Magnetic Sensor Array for Wirelessly Tracking Magnet Position and Orientation", IEEE Sensors Journal, vol. 10, no. 5, pp. 903-913, April 2010, doi: 10.1109/JSEN.2009.2035711

[23] Q. Shi, M. Wang, S. Song, M. Q.-H. Meng, "An Improved Simultaneously Magnetic Actuation and Localization Method based on Magnetic Sensor Array", 2019 IEEE SENSORS, Montreal, Canada, 2019, pp. 1-4, doi: 10.1109/SENSORS43011.2019.8956945

[24] M. Zhang, L. Yang, C. Zhang, Z. Yang, L. Zhang, "A 5-D Large-Workspace Magnetic Localization and Actuation System Based on an EYE-in-Hand Magnetic Sensor Array and Mobile Coils", IEEE Transactions on Instrumentation and Measurement, vol. 72, pp. 1-11, Jan. 2023, doi: 10.1109/TIM.20233238693

[25] S. Su, H. Dai, S. Cheng, P. Lin, C. Hu, B. Lv, "A Robust Magnetic Tracking Approach Based on Graph Optimization", IEEE Transactions on Instrumentation and Measurement, vol. 69, no. 10, pp. 7933-7940, May. 2020, doi: 10.1109/TIM.2020.2986843

[26] J. T. Sherman, J. K. Lubkert, R. S. Popovic, M. R. DiSilvestro, "Characterization of a Novel Magnetic Tracking System", IEEE Transactions on Magnetics, vol. 43, no. 6, pp. 2725-2727, May. 2007, doi: 10.1109/TMAG.2007.893314

[27] S. Guitron, A. Guha, S. Li, D. Rus, "Autonomous locomotion of a miniature, untethered origami robot using hall effect sensor-based magnetic localization", 2017 IEEE International Conference on Robotics and Automation (ICRA), Singapore, pp. 4807-4813, 2017, doi: 10.1109/ICRA.2017.7989560

[28] D. Cichon, R. Psiuk, H. Bauer, H. Töpfer, "A Hall-Sensor-Based Localization Method With Six Degrees of Freedom Using Unscented Kalman Filter", IEEE Sensors Journal, vol. 19, no. 7, pp. 2509-2516, Jan. 2019, doi: 10.1109/JSEN.2018.2887299

[29] G. Géron, C. Prelle, H. Al Hajjar, J. Terrien, M. U. Khan, "Characterization of a magnetic localization method based on Hall effect sensor array for microrobot position tracking", Journal of Micro-Bio Robotics, pp. 1-13, Mar. 2023, doi: 10.1007/s12213-023-00154-9
[30] E. P. Furlani, "Permanent Magnet Applications", *Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications*, ed. New York, USA, 2001, ch. 4, pp. 207-333, doi: 10.1016/B978-0-12-269951-1.X5000-1

[31] https://www.ti.com/lit/ds/symlink/tmag5170.pdf?ts=1701692783822&ref_url=https%253A%252F%252Fwww.ti.com%252Fproduct%252FTMAG5170

<figure><img src="image_14.png" /></figure>

**Gabriel Géron** received the engineering degree in mechanical engineering and the master degree in mechatronics systems from the Université de Technologie de Compiègne, Compiègne, France, in 2020. He is currently a PhD student in mechanical engineering with specialization in mechatronic, photonic and systems control with the Roberval

Laboratory of Université de Technologie de Compiègne.

His research interests include micro-mechatronic systems, micro-sensors, magnetic localization and tracking, optical localization and tracking, opto-mechatronics and numerical analysis.

<figure><img src="image_15.png" /></figure>

**Jérémy Terrien** received the Ph.D. degree in biomedical engineering from Université de technologie de Compiègne (UTC), Compiègne, France, in 2005. He is presently working as a Research Engineer in the department of electronics of the UTC.

His research interest focuses on nonstationary methods, time-frequency analysis, instrumentation, automation, sensors and digital actuators.

<figure><img src="image_16.png" /></figure>

**Muneeb Ullah Khan** received the double Ph.D. degrees in mechanical engineering with the specialization in microsystem design from Université de Technologie de Compiègne, France, and the Institute for Microtechnology, Technische Universität Braunschweig, Germany, in 2014. He is working as a Research Engineer with the Roberval Laboratory, Department of

Mechanical and Mechatronics Engineering, Université de Technologie de Compiègne.

His research interests include micro-mechatronic system designs, micro-actuators, micro-sensors, and nanofabrication technology.

<figure><img src="image_17.png" /></figure>

**Hani Al Hajjar** received the Ph.D. degree in photonics and telecommunications from Telecom Bretagne (Institut Mines-Telecom), Brest, France, in 2013. Since 2015, he has been an Associate Professor of Mechatronics and Communications Between Systems with the Roberval Laboratory, Université de Technologie de Compiègne, Compiègne, France. He is a

member of the Laboratory of Excellence Control of Technological Systems of Systems (labex MS2T).

His research interests include optical wireless communications, indoor access networks, high resolution optical localization and tracking, micro-mechatronic systems, opto-mechatronics, optical wireless actuation, and signal processing.

<figure><img src="image_18.png" /></figure>

**Christine Prelle** received the Ph.D. degree in industrial automatic control from Institut National des Sciences Appliquées Lyon, France, in 1997. She is currently a Professor of Control Engineering and Mechatronics with the Roberval Laboratory, Université de Technologie de Compiègne, France. She is also in-charge of the Doctoral School of Science for

Engineers, Université de Technologie de Compiègne.

Her research interest includes micro-mechatronics and control.