# Characterization of a magnetic localization method based on Hall effect sensor array for microrobot position tracking

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

Ullah Khan

► To cite this version:

Gabriel Géron, Christine Prelle, Hani Al Hajjar, Jérémy Terrien, Muneeb Ullah Khan. Characterization of a magnetic localization method based on Hall effect sensor array for microrobot position tracking. Journal of Micro-Bio Robotics, 2023, 10.1007/s12213-023-00154-9. hal-04066489

HAL Id: hal-04066489

https://utc.hal.science/hal-04066489v1

Submitted on 18 Apr 2025

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# Characterization of a magnetic localization method based on Hall effect sensor array for microrobot position tracking

Gabriel Géron<fnref n="1" /> · Christine Prelle<fnref n="1" /> · Hani Al Hajjar<fnref n="1" /> · Jérémy Terrien<fnref n="2" /> · Muneeb Ullah Khan<fnref n="1" />

## Abstract

This article presents a 2D measurement system using Hall effect sensor array for micro robotic applications. This system uses a trilateration method to track a microrobot equipped with a permanent magnet. The objective of this work is to characterize the impact of different physical parameters (i.e., sensors density, number of measurements, etc.) on the localization errors. The simulation and experimental results have shown a good correlation for the measurement system. The tracking trajectory of a mobile permanent magnet in xy-plane was successfully measured with a mean error of 43.9 ± 27.8 µm and of 47.5 ± 28.0 µm along x- and y-axis, respectively over a trajectory in a 3×3 mm<fnref n="2" /> workspace.

**Keywords** Sensor array · Trilateration · Hall effect sensors · 2D localization · Permanent magnet

## 1 Introduction

For several years, magnetically actuated microrobots have been actively developed to realize micromanipulation tasks and/-or to ensure 2D locomotion inside a workspace. As a result, these magnetic devices are more and more present in research work requiring small-scale actuation due to their great design flexibility. With the good understanding of magnetic and other physical phenomena at small scale and the rise of the miniaturized electronic components, it may offer a lot of possibilities in the design of the magnetic actuation system for the microrobot. These actuation systems use

magnetic effects to generate forces to move their magnetic part (e.g., permanent magnet) integrated inside the micro-robot. Among these actuation systems, we can find electro-magnetic devices using coils [1–3] or permanent magnets [4, 5]. As different applications, we can cite micro factories where magnetic microrobots are widely used to realize 2D conveying system of small parts [6, 7], and in lab-on-chip where they are used to carry out 2D cellular transportation [8–10]. In these applications, magnetic microrobots are of great interest to move and manipulate dynamically in-plane small objects along large travel range.

To achieve these tasks with high precision, the microrobots need to be controlled using high-resolution localization devices such as high-resolution cameras, laser-based tracking, etc. These devices can track the microrobot itself or one part of the microrobot with high accuracy. Typically, the vision-based measurement approach is one of the most used solutions to reach this high accuracy in applications such as micro-conveyance, cell micromanipulation, etc. [7–12]. This approach can deliver measurement data with micrometric resolution using cameras or microscopes to provide a feedback for a closed-loop control of the microrobot. Moreover, it exists other vision-based systems reaching nanometric, even sub-nanometric, resolution which can be adapted for these kinds of applications. For example, Cheng et al. [13] developed an ultraprecise three-axis visual motion tracking of microscale objects using a microscope and an interferometer with a real-time image

<figure><img src="image_2.png" /></figure>

Gabriel Géron
gabriel.geron@utc.fr

Christine Prelle
christine.prelle@utc.fr

Hani Al Hajjar
hani.al-hajjar@utc.fr

Jérémy Terrien
jeremy.terrien@utc.fr

Muneeb Ullah Khan
muneeb-ullah.khan@utc.fr

<footnote n="1">¹ Université de Technologie de Compiègne, Roberval (Mechanics, Energy and Electricity), CS 60319 – 60203, Compiègne Cedex, France</footnote>

<footnote n="2">² Department of Electronics, Université de Technologie de Compiègne, CS 60319 – 60203, Compiègne Cedex, France</footnote>
processing algorithm (measurement resolution < 1 nm). André et al. [14] used a CMOS camera in order to obtain x-, y- and θ-position measurements of a micro encoded target fixed on a robot by applying an image processing method (few nm for the in-plane positions and few µrad for the orientation).

The good resolution of the vision method is often limited to small displacements ranges (few micrometers). When the workspace of the microrobot becomes larger, the field of vision needs to be increased to completely cover the working area, which leads to the deterioration of the measurement resolution. In this scenario, vision-based approaches are no longer effective for precise positioning tasks over large workspace. Potential solutions can be applied to overcome this problem such as the use of multiple cameras covering the entire surface, but it can be complex to implement in a compact space and/or expensive. Otherwise, other means of measurement exist to address the duality high resolution/wide range measurement like optical tracking systems [15] or the sensor array approach [16]. The optical tracking systems provide good performance with resolution lower than 100 µm for large working areas compared to the vision-based systems. The problem of this type of solution is that optical link needs a direct view between the source and the microrobot. The sensor array approach seems more appealing since it addresses both, the high resolution and the wide range measurement solutions with a good integration ability.

In the context of localization, there is a wide variety of sensors such as photodiodes [17, 18], capacitive sensors [19, 20] or RFID sensors [21, 22], that can be used in an array configuration. The duality high resolution/wide range measurement is addressed because the sensor distribution can cover a very large working area without affecting the measurement resolution. Thanks to the sensor array disposition, this latter only depends on the sensor performance and on the selected localization algorithm. Among the many sensor technologies used in array, the Hall effect sensors have been

often used for localization in magnetic applications due to their easy implementation in comparison to types of sensors.

Several researchers developed Hall effect sensor arrays in various contexts. For example, Zhang et al. [23] presented a 2D localization method using 25 sensors in a sensing probe to localize a magnetic robot (Ø 250 mm × 150 mm magnet) in a large workspace. The xy-position of the robot is calculated with the help of an optimization algorithm, which minimizes the error between the magnetic simulation results and the magnetic measurements. With this approach, the localization errors were not sub-millimetric (less than 2 mm on average). Cichon et al. [24] are one of the few to obtain sub-millimetric errors with a localization method using Hall effect sensor array. They used an unscented Kalman filter to calculate the position along with the orientation of a magnet. Their results represent a maximum error of 71 µm for the position and 1.4 degrees for the orientation. These results were obtained using a very dense sensor array (distance between sensors = 2.4 mm) with a total of 36 sensors.

In this paper, we propose the study of a 2D magnetic localization system with a Hall effect sensor array using a trilateration method to track a permanent magnet (PM) integrated in a microrobot. For example, the microrobot to localize could be a planar electromagnetic conveyor moving at two degrees of freedom (x and y) above an electromagnetic surface, which actuates the microrobot [7]. A platform fixed on the conveyor would make it possible to carry micro-objects (e.g., a cell, a drug, etc.) in a large workspace (see Fig. 1). For the localization part, a Hall effect sensor array surface would be integrated below the actuation surface in order to determine the 2D position of a permanent magnet placed at the center of the conveyor. The localization and actuation would create a smart electromagnetic surface ensuring the closed-loop control of the microrobot to develop an automated micro-objects transportation system. The objective of the study presented in this paper is to design an optimal and realistic Hall effect sensor array to

<figure><img src="image_3.png" /><figcaption>**Fig. 1** Application example of a 2D magnetic localization with an integrated Hall effect sensor array for an electromagnetically actuated microrobot in the context of drug or cell transportation. The tracked</figcaption></figure>

magnetic element is a cubic PM (in black) located to the center of the microrobot and used to determine the 2D position of the microrobot by the trilateration method
accurately localize a microrobot with the lowest possible sensor density.

The section II of this article presents the measurement system and its characterization steps. A simulation of this measurement system and an analysis of the achieved results will be discussed in the section III. Finally, the section IV will show an experimental setup with experimental results to validate the simulation.

## 2 Hall effect sensor array system

In this study, the used conveyor has two orthogonal planar degrees of freedom, which makes the rotation movement not possible in this case. So, the measurement system is limited to the determination of the 2D (x,y) position. Nevertheless, in the case where we would like to determine this third degree of freedom (planar rotation), we could place at least two permanent magnets on the cross-shaped conveyor shown in Fig. 1. The planar rotation would be determined from the 2D position of the permanent magnets on the conveyor.

### 2.1 Working principle

The principle of the 2D magnetic localization system using a Hall effect sensor array is to calculate the 2D position of a mobile magnetic element after the collection and processing of magnetic field data of all sensors in the array. The relation between the magnetic field values and the position is provided by an algorithm adapted to the application. The choice of commercially available 3D Hall effect sensors (ALS31300) was done to give more information on the in-plane position of the magnetic element (i.e., a cuboid PM in our case) and therefore to obtain a more precise localization. These sensors provide the $B_x$, $B_y$- and $B_z$-components information of the surrounding magnetic field with a resolution of 0.1 mT in a wide magnetic range (-200 to 200 mT).

In order to carry out the simulation and to implement the localization algorithm, the mathematical model of the magnet's magnetic fields was established. According to [25], this model is derived from a Poisson's equation, including the magnetostatic field equations and reducing the cuboid magnet as a "magnetic charge" distribution. The model of the magnetic flux density is obtained by solving this Poisson's equation.

For a cuboid PM with length $L$, width $l$, height $h$ and magnetization $\vec{M}$, the $B_x$, $B_y$- and $B_z$-components of the magnetic field can be expressed as function of the position of the PM and the coordinates of the sensor's sensing point. In our case, we assume that the magnetization of the magnet is $\vec{M} = M\vec{z}$ and so the induction of the PM is $\vec{B} = B\vec{z}$ with $B = \mu_0 M$. Consequently, if $(x_M, y_M, z_M)$ are the coordinates of the magnet's center $(O_M)$ and $(x_S, y_S, z_S)$ are the coordinates of the sensor's sensing point $(O_S)$ in the reference frame $(O, x, y, z)$ (see Fig. 2), then the $B_x$, $B_y$- and $B_z$-components of the magnetic flux density can be expressed as following:

$$B_x = \frac{B}{4\pi} \sum_{k,n,m=1}^{2} (-1)^{k+n+m+1} \ln(y_S - y_m + R_{xyz}) \quad (1)$$

$$B_y = \frac{B}{4\pi} \sum_{k,n,m=1}^{2} (-1)^{k+n+m+1} \ln(x_S - x_n + R_{xyz}) \quad (2)$$

If $z_S - z_k \neq 0$,

$$B_z = \frac{B}{4\pi} \sum_{k,n,m=1}^{2} (-1)^{k+m+n} \text{atan}\left(\frac{(x_S - x_n)(y_S - y_m)}{(z_S - z_k)R_{xyz}}\right) \quad (3)$$

If $z_S - z_k = 0$,

$$B_z = \frac{\pi}{2} \quad (4)$$

With,

$$x_n = x_M + (-1)^n \frac{L}{2} \quad (5)$$

$$y_m = y_M + (-1)^m \frac{l}{2} \quad (6)$$

$$z_k = z_M + (-1)^k \frac{h}{2} \quad (7)$$

$$R_{xyz} = \sqrt{(x_S - x_n)^2 + (y_S - y_m)^2 + (z_S - z_k)^2} \quad (8)$$

<figure><img src="image_4.png" /><figcaption>Fig. 2 Cuboid magnet and sensor's sensing point in the (O,x,y,z) reference frame</figcaption></figure>
These expressions have been used to determine the position of the PM using a trilateration method detailed in the next paragraph.

## 2.2 Localization method: trilateration

In the context of magnetic localization, the classical trilateration method only uses 3 mono-axial Hall effect sensors [26] with the z orientation and forming an equilateral triangle of side d. Only the z-component of the magnetic flux density is recorded by the sensors. Indeed, for each sensor, the possible positions of the PM for a single value of the z-magnetic flux density would define a circle of equidistant potential position. The radius is determined from the linear approximation of the Bz-curve relative to the distance between the PM and the sensor (see Fig. 3c). Thus, the intersection of the three circles obtained by the three sensors would allow to estimate the unique position of the PM (see Fig. 3a). Nevertheless, all the possible positions of the PM for one value of the z-component of the magnetic flux density don't really define a circle because of the physical phenomenon expressed by the Eq. (3).

Moreover, the linear approximation of the Bz-curve adds measurement errors, namely when the PM is near or far from each sensor.

To solve this problem, we propose an improved version of the trilateration method using the three components of the magnetic flux density for the three sensors, using 3D Hall effect sensors (see Fig. 3b). For the i'th sensor, the x- and y-components Bxi and Byi will give an indication about the direction of the PM relative to the sensor. If we denote θi as the angle between the direction and the horizontal line for the i'th sensor, then it can be computed as following:

$$ \theta_i = \arctan\left(\frac{B_{yi}}{B_{xi}}\right) \qquad (9) $$

The z-component Bzi is then used in the direction defined by θi to determine the distance between the PM and the sensor i. Bzi can be expressed in function of this distance, denoting Ri for the i'th sensor:

$$ B_{zi}(R_i) = B_z(R_i \cos(\theta_i), R_i \sin(\theta_i), z_{M_i}, B, 0, 0, 0) \qquad (10) $$

<figure><img src="image_5.png" /><figcaption>**Fig. 3** Schematic layout of: (a) The principle of the classical trilateration method (b) The principle of the improved trilateration method when the PM is close to Sensor 1 (c) The shape of the Bz-component as function of the distance between the magnet's center and the sensor's sensing point R</figcaption></figure>
The value of $R_i$ is obtained by interpolating Eq. (10). Compared to the classical trilateration method, the Eq. (10) provides a better estimation of the distances $R_1, R_2$ and $R_3$, used for the calculation of the 2D position of the PM.

To calculate the position of the PM, the distances $R_1, R_2$ and $R_3$ of the sensors 1, 2 and 3 are used in 3 different cases. When the magnet is close to one sensor, we decided to use the two furthest sensors, knowing that we require only two sensors to calculate the 2D position of the magnet ($x_M, y_M$). This is done to avoid the low sensitivity area (i.e., area A of Fig. 3c) of $B_z$, when the magnet is close to one sensor, and to only work in the highly sensitive area (B) (see Fig. 3c). Moreover, the area (C) must be avoided because of near-zero sensitivity of $B_z$ and defines the maximum of the distance $d$ between the sensors.

If the magnet is close to the sensor 1,

$$y_M = \frac{2\sqrt{3\frac{R_2^2-R_3^2+d^2}{d}} - \sqrt{16R_2^2 - 4\frac{R_2^2-R_3^2+d^2}{d}}}{8} \quad (11)$$

$$x_M = \sqrt{3}y_M - \frac{R_2^2 - R_3^2}{d} \quad (12)$$

If the magnet is close to the sensor 2,

$$y_M = \frac{2\sqrt{3\frac{R_1^2-R_3^2+d^2}{d}} - \sqrt{16R_1^2 - 4\frac{R_1^2-R_3^2+d^2}{d}}}{8} \quad (13)$$

$$x_M = \frac{R_1^2 - R_3^2 + d^2}{d} - \sqrt{3}y_M \quad (14)$$

If the magnet is close to the sensor 3,

$$x_M = \frac{R_1^2 - R_2^2 + d^2}{2d} \quad (15)$$

$$y_M = \sqrt{R_1^2 - x_M^2} \quad (16)$$

The better estimation of the distance $R_1, R_2$ and $R_3$ and the non-consideration of the low sensitivity area should

improve the accuracy of the determined 2D position of the PM compared to classical trilateration method.

This localization method needs a triangular mesh for the Hall effect sensor array (see Fig. 4). So, in order to determine the absolute position of the magnet in this array, an assessment of the concentration of the magnetic flux density is done for all the triangular meshes. The trilateration method is used in the mesh having the highest magnetic concentration to obtain the relative position in this mesh and the absolute position in the array from the algorithm presented in the Fig. 5.

## 2.3 Impact of key design parameters on the localization efficiency

Besides the trilateration performance, the x- and y-localization errors depend on several parameters linked to the physical layout of the Hall effect sensor array (see Fig. 4), such as the magnet physical parameters and/or the sensor performance by itself. These parameters are listed as following:

* Size of the triangular mesh (d), defining the density of sensors in the array

* Elevation of the magnet relative to the sensor array (z)

* Physical parameters of the PM such as dimensions, magnetization, orientation, etc.

* Sensors' performances: noise level, linearity error, ...

In our case, a cubic shape PM has been chosen, nevertheless the trilateration method can be extended for different shapes (e.g., parallelepipedal or cylindrical magnet), knowing their magnetic model. Indeed, the magnetic models for each of these shapes will also have an impact on the localization performance.

In this study, after the selection of PM and sensor, our aim is to evaluate the impact of the sensors' density, the magnet position relative to the sensors and the noise level on the localization errors. In order to design an optimal Hall effect sensor array, a study of these three parameters was carried out through simulations.

<figure><img src="image_6.png" /><figcaption>Fig. 4 Hall effect sensor array surface with the cubic PM, magnetized in the z-direction, for a size of the triangular mesh equal to d and an elevation of the PM equal to z</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 5** Algorithm of the improved trilateration method. Three scenarios are differentiated depending on the proximity of the PM for each sensor</figcaption></figure>

# 3 Simulation

In this section, the study has been conducted in an elementary triangular mesh due to the symmetry of the sensor array. The use of the trilateration method implies that the localization errors are the same for all the triangular meshes in the sensor array. In addition, all simulation calculations have been realized for a cubic PM moving inside an equilateral

triangle formed by three sensors. The coordinates of the sensors are the same as depicted in the Fig. 2. The position of the Sensor 1 corresponds to the reference point.

## 3.1 Modeling of the sensor response

The localization algorithm requires the output data delivered by the 3D Hall effect sensors that corresponds to the three
components (i.e. $B_x$, $B_y$ and $B_z$) of the magnetic flux density. So, the magnetic field measurement $\overrightarrow{B_S}$, delivered by the $i^{th}$ sensors, is modeled using the expressions of the magnetic flux density defined by the Eqs. (1)-(4) with the addition of measurement noise and can be expressed as following:

$$ \overrightarrow{B_S} = \begin{pmatrix} B_x(x_M, y_M, z, x_{S_i}, y_{S_i}, 0) \\ B_y(x_M, y_M, z, x_{S_i}, y_{S_i}, 0) \\ B_z(x_M, y_M, z, x_{S_i}, y_{S_i}, 0) \end{pmatrix} + \begin{pmatrix} \epsilon_{x_i} \\ \epsilon_{y_i} \\ \epsilon_{z_i} \end{pmatrix} \quad (17) $$

where $(x_M, y_M)$ is the real in-plane position of the magnet center, $z$ is the elevation of the PM relative to the sensors, $(x_{S_i}, y_{S_i})$ are the coordinates of the $i^{th}$ sensors and $(\epsilon_{x_i}, \epsilon_{y_i}, \epsilon_{z_i})$ are random variables following a normal distribution with mean zero and standard deviation 1-sigma noise of the $i^{th}$ sensor.

## 3.2 Simulation results

The objective of the simulation is to understand the impact of the parameters on the localization errors in x- and y-axis using the trilateration method as discussed in the sub section II.C. In an elementary sensor array, the sensors are spaced with a distance $d$ and the magnet is elevated at an altitude $z$. An additional parameter can be taken into account: the number of successive measurements (denoted $N$) of the sensor output. Indeed, $N$ should have an impact on the measurement noise.

For the simulation, we have selected a $5 \times 5 \times 5$ mm³ cubic PM with a z-oriented magnetization equal to 1,43 T and a 1-sigma noise equal to 0.15 mT for the sensors. The localization errors are calculated for all the positions inside the triangle mesh with a step of 0.2 mm. These errors compare the x- and y-position of the PM calculated by the trilateration algorithm and the real x- and y-position. In order to observe separately the impact of each parameter on the mean localization error, two of these parameters have been fixed while the third one is varied.

Figure 6 shows the effect of $N$ for $d=6$ mm and $z=4$ mm, proving a degradation of the localization precision as the noise level decreases. Obviously, if the number of successive measurements $N$ increases, then the reading time of the output data of the sensors is also increased. In the next simulations and in the experiments, the value of $N$ is chosen to be 100 in order to keep a suitable reading time, while keeping an acceptable localization error.

A study of the impact of the $d$ and $z$ values in the precision of the measurement system was also carried out. In this study, $N$ has been set to 100 and only $d$ and $z$ are varying independently. First, the mesh size $d$ is fixed to 6 mm. We increased the elevation of the PM from 2 to 11 mm and calculated the mean value of the localization errors inside the mesh (see Fig. 7a). The mean localization error is minimum

<figure><img src="image_8.png" /><figcaption>Fig. 6 Effect of the noise level in the localization errors inside the triangular mesh with $d=6$ mm and $z=4$ mm</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7 Localization errors as function of (a) the elevation of the PM relative to the sensors with $d=6$ mm and $N=100$ (b) the size of the triangular mesh with $z=4$ mm and $N=100$</figcaption></figure>

for $z=4$ mm. This is mainly due to the sensitivity value of $B_z$-induction. As it can be seen in Fig. 8a, when the magnet is very far from the sensor, the magnetic sensitivity becomes lower and lower and therefore more and more sensitive to the noise. Moreover, when the magnet is very close to the plane of the sensor array, the part of the C area (see Fig. 3) is much more present and the part of the B area is more
<figure><img src="image_10.png" /><figcaption>**Fig. 8** (a) B<sub>z</sub>-magnetic induction curves as function of the elevation z with d=6 mm and N=100 (b) Definition of the part of B<sub>z</sub>-curve which are used for the improved trilateration method relative to the distance between sensors with z=4 mm and N=100</figcaption></figure>

shortened. This can lead to a predominance of the low sen-
sitivity areas (A and C) compared to the high sensitivity
area (B), hence an increase of the localization accuracy for
the case z=2 mm.

In the next simulation, z is fixed to 4 mm and we varied
the mesh size *d* from 3 to 11 mm (see Fig. 7b). We observed
that the localization errors decrease when the size of the
triangular mesh decreases too. These observations are due
to the use of the low sensitivity areas (A and C) when the
mesh size becomes larger. Thus, the larger the mesh size,
the more the A or C areas are present in the part of B<sub>z</sub>-
curve used for the trilateration algorithm and the more the
measurement noise will have an impact on the localization
errors. For example, in Fig. 8b, when the mesh size is equal
to 3 mm, the part of the useful B<sub>z</sub>-curve (in the black rectan-
gle) defines a high magnetic sensitivity reducing the impact
of the measurement noise. However, when the mesh size is
equal to 11 mm, the part of the useful B<sub>z</sub>-curve (in the green
rectangle) defines a lower magnetic sensitivity highlighting

the influence of the measurement noise on the localization
efficiency. On the other hand, the constraints imposed by the
footprint of each sensor will limit the integration avoiding
very small values for *d*. Besides these constraints, the small
values for *d* implies an increase of the cost of the sensor
array design with a large number of sensors to finally obtain
a small improvement of the localization precision. Consider-
ing also all these reasons, a value of the parameter *d* equal
to 6 mm is chosen to have a good compromise between the
lowest possible precision and the limit distance between two
successive sensors (equal to 6 mm for the chosen sensors).

So, the appropriate disposition for the sensor array con-
sidering the chosen PM is as follow: d=6 mm, z=4 mm and
N=100. For this configuration, the theoretical localization
error is 3.9 ± 3.4 µm in x-axis and 3.2 ± 2.6 µm in y-axis.

**4 Experimental setup and results**

In this section, the experimental setup that was developed
is shown and the experimentation that were conducted to
validate the simulation results are presented.

**4.1 Experimental setup**

The experimental setup is composed of Printed Circuit
Board (PCB) with nine 3D digital Hall effect sensors, con-
trolled by a microcontroller (an LPC804 from NXP) based
on an Arm Cortex-M0+ . These sensors provide the three
components of the magnetic flux density with a 1-sigma
measurement noise equal to 0.15 mT. They were distributed
so as to make three elementary triangular meshes of different
sizes (6 mm, 16 mm and 26 mm) as shown in Fig. 9a. These
distributions help in studying the impact of the parameter d
on the localization error.

A 5×5×5 mm³ NdFeB cubic PM with a magnetization
of 1.43 T is moved over this PCB with the help of a three-
axis positioning system. This system consists of two motor-
ized stages (2×UMR8.25 + 2×TRA25CC from Newport)
that move the PM along x- and y-axis with a resolution of
0.15 µm. To obtain a given height on the z-axis parallel to
the PCB, a z-manual stage (UMR5.16 + BM11.16 from
Newport) is used with a resolution of 20 µm.

A Labview program has been developed to drive the two
motorized stages via a dedicated controller. The program
also allows to collect the digital data of the three magnetic
field components of each sensor (see Fig. 9b). The collected
digital data is averaged over N measurements per sensor in
order to increase the signal to-noise ratio. These measured
averaged data are processed in the localization algorithm in
order to determine the 2D relative position of the magnet
inside each triangular mesh at a fixed altitude.
<figure><img src="image_11.png" /><figcaption>**Fig. 9 (a)** Experimental prototype for the system validation with the three-axis positioning system and the sensors PCB with the three different triangular meshes (6 mm, 16 mm and 26 mm) (**b**) Block diagram of the principal control of the PM movement of the data collection</figcaption></figure>

## 4.2 Results

Firstly, an evaluation of the localization errors inside the smallest mesh ($d=6$ mm) with $z=4$ mm was carried out as function of the noise level, present in all three sensors. The results are presented in Fig. 10 showing obviously an improvement of the localization error when the number of measurement $N$ increases. However, a great number of measurement $N$ implies an increase of the reading time, thus limiting the motion dynamics of the PM for a desired precision. In Fig. 10, this reading time is determined relative to the reference reading time (RRT), corresponding of the experimental reading time for $N=1$. This RRT is defined by the difference between the moment when we launch the reading and the moment when we received the response of microcontroller. We experimentally estimated this value at 0.294 s. Moreover, the experimental results confirm the good tradeoff between a reasonable localization error and a

<figure><img src="image_12.png" /><figcaption>**Fig. 10** Experimental results of the x/y-axis localization errors and the reading of sensors relative to the noise level with $d=6$ mm and $z=4$ mm</figcaption></figure>
short reading time, obtained for N=100. This value of N is chosen for the configuration test of the triangular mesh for the sensor array.

Secondly, an evaluation of the localization errors was carried out for several values of the parameters *d* and *z* for N=100, defining several configurations of the triangular mesh. Only the two smaller meshes are considered, because the larger mesh (*d*=26 mm) does not give interesting measurement resolutions. Table 1 compares simulation and experimental localization errors for the configuration *d*=6 mm and *z*=4 mm.

The tracking ability of the measurement method has been tested with the magnet following a particular trajectory inside the smallest mesh with a step size of 0.1 mm (see Fig. 11). During this test, the PM started at the position (1; 0) and traveled a distance of 1.5 mm along the y-axis. Then, the PM traveled a distance of 3 mm along the x-axis and finally a distance of 1.5 mm along y-axis until reaching the position (4; 3). At each step of 0.1 mm, the xy-position of the PM was calculated with the trilateration method and the localization errors along x- and y-axis were estimated compared to the real position of the PM. The results show the measurement system can track the PM with a mean error of 43.9 ± 27.8 µm in x-axis and of 47.5 ± 28.0 µm in y-axis. These errors are slightly higher than in Table 1, because the results in Table 1 were obtained by an averaging over the whole mesh.

Figure 12 shows the obtained localization errors for each point of the magnet trajectory. In these graphs, we can notice some error steps located at specific points. These latter correspond to a transition between two couple of selected sensors by the trilateration algorithm used for the position calculation, according to the relative position of the magnet with each sensor. The manufacturing uncertainty of each sensor could also explain these results because, in this case, the sensors don't provide exactly the same values of the PM's magnetic inductions. The difference between the sensors has an impact directly on the localization errors which are different as function of the proximity of each sensor.

## 4.3 Discussion

Different methods using planar sensor array were developed by researchers. Table 2 lists some planar sensor array

<table><caption>**Table 1** X/Y-axis errors (in µm) for different configurations of the triangular mesh for N=100</caption><thead><tr><th rowspan="2"></th><th colspan="2">z=4 mm</th><th colspan="2">z=10 mm</th></tr><tr><th>Simulation</th><th>Experiment</th><th>Simulation</th><th>Experiment</th></tr></thead><tbody><tr><td>d=6 mm</td><td>3.215</td><td>30.51</td><td>26.68</td><td>41.82</td></tr><tr><td>d=16 mm</td><td>137.6</td><td>406.6</td><td>59.08</td><td>123.9</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 11** Tracking of the PM inside a triangular mesh with d=6 mm, z=4 mm and N=100</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 12** Evaluation of the tracking errors for a given setpoint of the magnet position along (a) the x-axis (b) the y-axis</figcaption></figure>
<table><caption>**Table 2** Experimental performance of our system and other published work about planar sensor array</caption><thead><tr><th>System</th><th>Sensor type</th><th>Localization error (mm)</th><th>Target dimensions</th><th>Sensor density (number of sensors/mm²)</th></tr></thead><tbody><tr><td>[17]</td><td>Photodiode</td><td>1.3</td><td>7.5×4.5 mm²</td><td>0.44</td></tr><tr><td>[19]</td><td>Capacitive</td><td>0.02</td><td>Ø 0.05 mm</td><td>2500</td></tr><tr><td>[23]</td><td>Hall effect sensor</td><td>&lt;2</td><td>5×5×5 mm³</td><td>0.02</td></tr><tr><td>[24]</td><td>Hall effect sensor</td><td>0.071</td><td>8×4×3 mm³</td><td>0.1</td></tr><tr><td>[26]</td><td>Hall effect sensor</td><td>0.995</td><td>5×5×5 mm³</td><td>0.01</td></tr><tr><td>Our system</td><td>Hall effect sensor</td><td>0.047</td><td>5×5×5 mm³</td><td>0.08</td></tr></tbody></table>

devices using different sensor types and compares it relatively to sensor type, localization error, target dimensions and sensor density. For example, in [19], small bioparticles can be detected by using a CMOS capacitive sensor array, with a low resolution of 20 µm and a high sensor density equal to 2500 sensors/mm². Conversely, in [23], a magnetic robot is localized by using a Hall effect sensor array, with a localization error less than 2 mm and a low sensor density equal to 0.02 sensors/mm². Compared to these systems, our method reveals a good compromise in order to localize with high precision a cubic magnet with the lowest sensor density, despite the experimental conditions.

These gaps between theoretical and experimental results could be explained by the existence of assembly errors in the experimental prototype and some variations in the sensor performances. A meticulous placement of the PM and the sensors has been carried out to minimize them. However, it is difficult to eliminate them. Among these uncertainties, we can cite: the specific characteristics of the sensor (gain, offset, non-linearity), the orientation of the PM relative to the PCB plan and the position error of the sensors' sensitive point (see Fig. 13). Indeed, the non-linearity of each

sensor, as their gain and their offset, give wrong data and wrong information about the behavior of real magnetic fields, implying a change of the latter at the sensor output. Moreover, the value of magnetic fields at a given point differs when changing the orientation of the PM with respect to the plane of the sensor. That's why, the chosen analytical model for the trilateration algorithm doesn't match exactly to the behavior of real magnetic fields and that implies a deterioration of the measurement accuracy. Concerning the sensors' sensitive point, a position offset of this point also causes an offset of the magnetic curves and therefore an offset on the calculated position.

In order to highlight this impact, these different defaults have to be considered in the mathematical model of the magnetic inductions $B_x$, $B_y$ and $B_z$ used to model the sensor output in the simulation. The aim is to prove that these issues have a real impact in the measurement precision and match well to the experimental values of the Table 1. For each sensor, they can be modeled by mathematical expression depending on several parameters and so they can be included in the mathematical model of the magnetic induction. The estimation of these parameters can be carried out

<figure><img src="image_15.png" /><figcaption>**Fig. 13** Explanatory diagrams of the various defaults justify the differences between theoretical and experimental results on the localization errors. The corresponding analytical expressions of these differ-</figcaption></figure>

ent issues, expressed under each of the diagram, are integrated in the analytical model to best represent the real magnetic fields
<table><caption>**Table 3** X/Y-axis errors (in µm) for different configurations of the tri- angular mesh for N=100 with the consideration of the defaults</caption><thead><tr><th colspan="4" style="text-align: left;">Z=4 mm</th></tr><tr><th></th><th colspan="2" style="text-align: center;">Simulation</th><th style="text-align: center;">Experiment</th></tr><tr><th></th><th style="text-align: center;">No calibration</th><th style="text-align: center;">Calibration</th><th></th></tr></thead><tbody><tr><th scope="row" style="text-align: left;">d=6 mm</th><td style="text-align: center;">3.215</td><td style="text-align: center;">29.95</td><td style="text-align: center;">30.51</td></tr><tr><th scope="row" style="text-align: left;">d=16 mm</th><td style="text-align: center;">137.6</td><td style="text-align: center;">414.1</td><td style="text-align: center;">406.6</td></tr><tr><th colspan="4" style="text-align: left;">Z=10 mm</th></tr><tr><th></th><th colspan="2" style="text-align: center;">Simulation</th><th style="text-align: center;">Experiment</th></tr><tr><th></th><th style="text-align: center;">No calibration</th><th style="text-align: center;">Calibration</th><th></th></tr><tr><th scope="row" style="text-align: left;">d=6 mm</th><td style="text-align: center;">26.68</td><td style="text-align: center;">42.04</td><td style="text-align: center;">41.82</td></tr><tr><th scope="row" style="text-align: left;">d=16 mm</th><td style="text-align: center;">59.08</td><td style="text-align: center;">124.3</td><td style="text-align: center;">123.9</td></tr></tbody></table>

by the calibration of each sensor relative to the new modeling of the real magnetic induction. By incorporating these estimated parameters in the mathematical model of the simulated sensor measurement for the trilateration algorithm, we succeeded in shifting the theoretical errors towards the experimental errors (see Table 3).

In the simulation, the errors with the calibration are more significant than the errors without calibration because of the error accumulation induced by the consideration and the addition of the defaults in the magnetic field measurement model. Thus, the errors induced by the behavior of the sensors, the orientation of the PM and the offset of the sensors' sensitive point position are added to the error induced by the measurement noise of the sensors and the final results have to approach the experimental values given by Table 1. These results show that the precision of the measurement system by the trilateration method is very sensitive considering the assembly uncertainty and sensor performances. In the future, we will improve the placement of the magnet and the sensors with using advance assembly equipment during setup. Furthermore, we will also include the assembly errors in the trilateration algorithm to compensate their effect on the localization error values.

## 5 Conclusion

In this paper, we have characterized a 2D magnetic localization system based on the use of 3D Hall effect sensor array with high precision and large 2D workspace. This system calculates the 2D position of a cubic magnet, integrated in a microrobot, at a fixed elevation with a modified version of the trilateration method. The used trilateration algorithm and the meticulous placement of the PM and the sensors allow to have a better estimation of the 2D position of the magnet compared to the classical trilateration method. Indeed, the theoretical and experimental results have allowed to achieve micrometric errors and presented an optimization of the disposition of the sensor array and of the acquisition time of

each sensor. This localization technique can be suited for any 2D motion robots to which a cubic PM has been attached without any magnetic disturbance in the surroundings.

In perspective, to improve the operation of the measuring system, the PM orientation, the sensors' non-linearity and the sensitive point position issues will be considered in the model. Moreover, we intend to find and test other localization method (Newton-Raphson, Kalman filter...) and to compare their effectiveness in order to select the best method for a precise measure with the lowest sensor density.

**Acknowledgements** 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.

**Author's contributions** All authors participated on the definition and study of the method presented in this article. G.G. developed the theoretical model, carried out the simulations and performed the experiments. G.G., J.T. and M.U.K. contributed to the design of the experimental setup. All authors participated in the writing of this manuscript. G.G. wrote the main manuscript text and made the figures and tables. C.P., H.A.H., J.T. and M.U.K. revised the manuscript.

**Funding** 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.

**Availability of data and materials** This declaration is not applicable.

## Declarations

**Competing interests** All authors have no competing interests that might be perceived to influence the results and/or discussion reported in this paper.

## References

1. Kim J, Park C, Yoo J, Kim S-J (2013) Two-dimensional motion control of micro-robot by using two pairs of Helmholtz coils. 2013 IEEE International Symposium on Robotics (ISR 2013)

2. Chowdhury S, Johnson BV, Jing W, Cappelleri DJ (2017) Designing local magnetic fields and path planning for independent actuation of multiple mobile microrobots. J Micro-Bio Robot 12:21-31

3. Kantaros Y, Johnson BV, Chowdhury S, Cappelleri DJ, Zavlanos MM (2018) Control of magnetic microrobot teams for temporal micromanipulation tasks. IEEE Trans Robot 34(6):1472-1489

4. Zarrouk A, Belharet K, Tahri O, Ferreira A (2019) A Four-magnet System for 2D Wireless Open-Loop Control of Microrobots, 2019 International Conference on Robotics and Automation (ICRA)

5. Yousefi M, Pishkenari HN (2021) Independent position control of two identical magnetic microrobots in a plane using rotating permanent magnets. J Micro-Bio Robot 17:59-67

6. Komori M, Hirakawa T (2005) A magnetically driven linear microactuator with new driving method. IEEE/ASME Trans Mechatronics 10(3):335-338

7. Arora N, Khan MU, Petit L, Lamarque F, Prelle C (2019) Design and development of a planar electromagnetic conveyor for the microfactory. IEEE/ASME Trans Mechatronics 24(4):1723-1731
8. Go G, Kwak D, Piao L, Choi H, Jeong S, Lee C, Park BJ, Ko SY, Park J-O, Park S (2013) Manipulation of micro-particles using a magnetically actuated microrobot. Mechatronics 23(8):1037–1043

9. Jing W, Chowdhury S, Guix M, Wang J, An Z, Johnson BV, Cap-pelleri DJ (2019) A microforce-sensing mobile microrobot for automated micromanipulation tasks. IEEE Trans Autom Sci Eng 16(2):518–530

10. Beyeler F, Neild A, Oberti S, Bell DJ, Sun Y, Dual J, Nelson BJ (2007) Monolithically fabricated microgripper with integrated force sensor for manipulating microobjects and biological cells aligned in an ultrasonic field. J Microelectromech Syst 16(1):7–15

11. Zhou L, Chapuis Y-A, Fukuta Y, Mita Y, Braun F, Fujita H (2005) Architecture and implementation of distributed control system for MEMS-based intelligent motion surface. Proceedings of the IEEE International Symposium on Industrial Electronics, 2005. ISIE 2005

12. Zarrouk A, Belharet K, Tahri O (2020) Vision-based magnetic actuator positioning for wireless control of microrobots. Robot Auton Syst 124:103366

13. Cheng P, Menq C-H (2017) Ultraprecise Three-axis visual motion tracking microscopic objects. IEEE Trans Instrum Meas 66(10):2597–2605

14. André AN, Sandoz P, Mauzé B, Jacquot M, Laurent GJ (2020) Sensing one nanometer over ten centimeters: a microencoded tar-get for visual in-plane position measurement. IEEE/ASME Trans Mechatronics 25(3):1193–1201

15. Malak S, Al Hajjar H, Dupont E, Khan MU, Prelle C, Lamarque F (2021) Optical localization and tracking method of a mobile micro-conveyor over a smart surface. IEEE Sensors J 21(9):10618–10627

16. Johnson KJ, Rose-Pehrsson SL (2015) Sensor array design for complex sensing tasks. Annu Rev Anal Chem 8:287–310

17. Ataka M, Mita M, Fugita H (2007) The layer-built sensor/actuator integrated array for the 2D feedback conveyance. 2007 IEEE 20th International Conference on Micro Electro Mechanical Systems (MEMS)

18. Ivan IA, Ardeleanu M, Laurent GJ, Tan N, Clevy C (2012) The metrology and applications of PSD (position sensitive detector) sensors for microrobotics. 2012 International Symposium on Optomechatronic Technologies (ISOT 2012)

19. Romani A, Manaresi N, Marzocchi L, Medoro G, Leonardi A, Altomare L, Tartagni M, Guerrieri R (2004) Capacitive sensor

array for localization of bioparticles in CMOS lab-on-a-chip. 2004 IEEE International Solid-State Circuits Conference (IEEE Cat. No.04CH37519)

20. Sonar HA, Yuen MC, Kramer-Bottiglio R, Paik J (2018) An any-resolution pressure localization scheme using a soft capacitive sensor skin. 2018 IEEE International Conference on Soft Robotics (RoboSoft)

21. Hou J, Zhu Y, Zhang L, Fu Y, Zhao F, Yang L, Rong G (2009). 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

22. Martinelli F (2016) Robot localization using the phase of passive UHF-RFID signals under uncertain tag coordinates. J Intell Rob Syst 82:577–593

23. Zhang M, Yang L, Zhang C, Yang Z, Zhang L (2021) Simultaneous Actuation and Localization of Magnetic Robots Using Coils and Eye-In-Hand Hall-Effect Sensors. 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)

24. Cichon D, Psiuk R, Brauer H, Töpfer H (2019) A Hall-sensor-based localization method with six degrees of freedom using unscented kalman filter. IEEE Sensors J 19(7):2509–2516

25. Furlani EP (2001) Chapter 4: Permanent Magnet Applications. Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications, pp. 207–333

26. Guitron S, Guha A, Li S, Rus D (2017) Autonomous locomotion of a miniature, untethered origami robot using hall effect sensor-based magnetic localization. 2017 IEEE International Conference on Robotics and Automation (ICRA)

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