# Two-Dimensional Digital Electromagnetic
Micro-Conveyance Device

Célien Bergeron, Gabriel Géron, Laurent Petit, Erwan Dupont, Nicolas Piton,
Christine Prelle

► To cite this version:

Célien Bergeron, Gabriel Géron, Laurent Petit, Erwan Dupont, Nicolas Piton, et al. Two-Dimensional Digital Electromagnetic Micro-Conveyance Device. Actuators, 2026, 15 (2), pp.75. 10.3390/act15020075. [https://doi.org/10.3390/act15020075](https://doi.org/10.3390/act15020075)

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Article

# Two-Dimensional Digital Electromagnetic Micro-Conveyance Device

Célien Bergeron ID, Gabriel Géron ID, Laurent Petit *, Erwan Dupont ID, Nicolas Piton and Christine Prelle ID

Laboratoire Roberval, Université de technologie de Compiègne, CS 60319-60203 Compiègne Cedex, France;
celien.bergeron@utc.fr (C.B.); gabriel.geron@utc.fr (G.G.); erwan.dupont@utc.fr (E.D.); nicolas.piton@utc.fr (N.P.)

* Correspondence: laurent.petit@utc.fr

## Abstract

This paper presents a 2D micro-conveyance device based on a 3 × 3 electromagnetic digital actuator array. This device allows the conveyed object to be moved between several discrete positions distributed in the xy-plane through a collaborative actuation of the digital actuators. Each digital actuator includes a mobile permanent magnet placed in a square cavity and can be moved between four discrete positions. An analytical model of the digital actuators was proposed and used to design the conveyance device. Then, a prototype was built using rapid prototyping techniques and was experimentally characterized. The reachable workspace of the conveyance device is 56 mm × 56 mm in the xy-plane, and the proposed architecture enables the workspace to be easily enlarged by adding elementary modules. The distance between two discrete positions is 4 mm, and the positioning repeatability was measured as 5.5 µm. The maximum conveyance velocity and transportable mass were found to be up to 16 mm.s⁻¹ and 15 g, respectively.

**Keywords:** micro-conveyance device; planar motion; actuator array; digital actuators; electromagnetic actuator

## 1. Introduction

Nowadays, many products offer a large number of functions in a reduced volume, which leads to the integration of an increasing number of sensors and actuators. The manufacturing of such products represents a challenge, especially when considering the environmental impact and particularly the energy consumption of the manufacturing process. The microfactory concept provides an answer to this issue and proposes the manufacturing of micro or miniature products using manufacturing tools with dimensions adapted to those of the produced objects [1]. Microfactories have to present high levels of flexibility and modularity in order to be easily adapted to different manufactured products [2]. A high level of automation is required, and this is particularly the case for the conveyance system, which represents a key component because it ensures the flow of parts within the microfactory.

In the literature, many micro-conveyance systems are described, and the term “micro” refers either to the characteristic dimensions of the structures or to the micrometric precision of the conveyance mechanism itself. These systems can be classified into three main categories corresponding to different types of contact with the conveyed objects: liquid-based, solid-based and air-jet-based. Liquid-based conveyance systems can manipulate cells or droplets [3,4] as well as micro-objects [5] using the electrowetting principle, but high-accuracy motion is not always possible and the motion is limited to predefined

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Received: 15 December 2025

Revised: 19 January 2026

Accepted: 20 January 2026

Published: 26 January 2026

**Copyright:** © 2026 by the authors.
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This article is an open access article
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paths [6]. With solid-based conveyance systems, there is a mechanical contact between the conveyor and the conveyed objects and devices using thermal actuators [7], electromagnetic actuators [8–10], piezoelectric actuators [11,12], electrostatic actuators [13,14], or dielectric elastomer actuators [15] have been presented. The properties and performance of the obtained motions are strongly linked to the considered actuation principles. With air-jet conveyance systems, the conveyed objects are levitated and moved by pneumatic airflow [16–18] or magnetic levitation [8]. The main interest of these air-jet solutions is the contact-free conveyance adapted to fragile objects; nevertheless, their control remains generally complex [19].

Most of these systems are composed of several actuators or modules that enable coverage over large surfaces and ensure the required conveyance flexibility. The elementary modules or actuators can generate either an analog action [11,13] or a digital action [7,10]. Analog actuators enable us to reach any positions in their working stroke limits, thus enabling continuous motions [13], and are generally controlled in closed loop [20,21], which enables compensation for disturbances. Digital actuators are composed of a mobile part that can switch between a limited number of well-known discrete positions [20,22,23]. With digital actuators, energy supply is only needed for a position change, and no energy supply is needed to hold the mobile part in a discrete position even in the presence of small disturbances [24,25]. Due to the high positioning repeatability of the discrete positions, the control of digital actuators is implemented in an open loop with simple control signals (i.e., pulse signals) [26] without the need of added sensors that facilitate their integration [25].

Compared with closed-loop control actuators, the control of digital actuators does not allow compensation for manufacturing errors, so high manufacturing quality is generally required for these actuators. However, to realize complex tasks, such as a conveyance with several degrees of freedom over a large surface, a collaborative actuation principle is generally used to generate the motion and reach an increased number of positions.

In this paper, a micro-conveyance device based on a digital actuator array is presented. The main originality of the proposed work lies in the unique architecture of the device, which enables discrete motions in two orthogonal directions and offers a high level of modularity by allowing easy arrangement to cover the desired surface. In previous work, this kind of actuator has been used to realize a planar conveyance device based on a stick-slip principle [22,23] and it has been shown that the conveyed distance can be adapted according to the control parameters. The main drawback of using the stick-slip principle is its friction dependence, which makes it difficult to accurately predict the conveyed object displacement when the transported mass varies. The work presented in this paper focuses on an alternative solution whose main originality lies in the digital nature of the generated two-dimensional motion. The main interest is that the conveyed object can reach discrete positions distributed in the plane and the conveyance distance is no longer dependent on friction conditions or transportable mass.

In Section 2, the principles of both actuator array and conveyance device are presented. The modeling of the device used during the design process is described in Section 3. Section 4 presents a prototype of the conveyance device and an experimental characterization. Finally, Section 5 concludes the paper and proposes some perspectives.

## 2. Actuator and Micro-Conveyance Principle

### 2.1. *Principles of the Micro-Conveyor and Actuator Array*

The proposed micro-conveyance device is based on several modules that can be assembled to cover a large surface and move small objects in the xy-plane. The objects to be conveyed are placed on mobile platforms that are moved by elementary micro actuators integrated in each module. Figure 1 shows a schematic view of such a conveyance
device with several assembled modules and two mobile platforms. This module-based
design allows the conveying surface to be defined in terms of both size and trajectory.
It also ensures that the conveying surface is limited to what is strictly required, thereby
avoiding unused areas, while enabling rapid reconfiguration of the conveying surface
through module rearrangement. This paper focuses on one module composed of a digital
actuator array. This array can move the mobile platform in two-dimensional motion using
a collaborative actuation principle.

<figure><img src="image_4.png" /><figcaption>**Figure 1.** Schematic view of the modular conveyance system (the arrows indicate the motion of the platform).</figcaption></figure>

The considered actuator array is composed of nine actuators arranged in a 3 × 3 matrix arrangement (Figure 2a). Each elementary actuator includes a mobile permanent magnet (MPM), placed inside a square cavity, which can be moved between the four corners of the cavity (Figure 2b). Four fixed permanent magnets (FPMs) are placed around the cavity in order to generate magnetic holding forces on the MPM. This holding force ensures that each MPM remains in one of its discrete stable positions, even in the presence of external disturbances and magnetic interactions with other MPMs; without FPMs, this holding force is not ensured and the MPMs may no longer remain in stable resting positions.

The switching of the MPM along the x and y axes is performed by means of a Printed
Circuit Board (PCB) placed below the cavity (Figure 2c). This PCB integrates two sets of
five perpendicular wires, connected in series. When a current is injected through a set of
wires, a Lorentz force is generated and ensures switching of the MPM. The wires orientated
along the x-axis ensure y-axis switching of the MPM and vice versa. The PCB is composed
of four layers, with the two top layers (L₁ and L₂) dedicated to the switching wires and
the two bottom layers (L₃ and L₄) for the routing. The y-axis switching wires are placed
in the first layer (L₁) of the PCB (distance between the bottom side of the MPM and the
y-axis switching wires called d₄), whereas the x-axis switching wires are placed in the
second layer (L₂) of the PCB (distance between the bottom side of the MPM and the x-axis
switching wires called d₅). The vertical distance between the two top PCB layers (L₁ and
L₂) is 100 µm, and the wire thickness is 35 µm. The distance (along the x- or y-axis) between
adjacent wires of the same layer is 250 µm. Moreover, a thin glass layer with a thickness of
170 µm is also placed between the PCB and the MPM to avoid electrical contact between
them and to provide a flat and homogeneous surface for the MPM displacement. The
geometrical parameters of the device and the properties of the PMs are given in Table 1.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Schematic views of the digital actuator array: (a) top view, (b) top view of an elementary actuator, and (c) side view (A-A) of an elementary actuator.</figcaption></figure>

An essential requirement for actuator arrays is the uniform behavior of all actuators in order to ensure robust control of such devices. In the proposed architecture, magnetic interactions between the PMs generate inhomogeneities in the magnetic forces. To obtain a homogeneous behavior for the nine elementary actuators, balancing permanent magnets are placed around the array to balance the magnetic holding forces particularly for the actuators located at the array boundaries (Figure 2a).
<table><caption>**Table 1.** Dimensional and magnetic properties of the actuator.</caption><thead><tr><th>Permanent Magnets</th><th>Dimensions (mm)</th><th>Magnetization (T)</th></tr></thead><tbody><tr><td>MPM</td><td>10 × 10 × 3</td><td>1.31</td></tr><tr><td>FPM</td><td>2 × 2 × 2</td><td>1.17</td></tr><tr><td>Balancing magnet</td><td>2 × 2 × 2</td><td>1.17</td></tr><tr><td colspan="3">Masses (g)</td></tr><tr><td>MPM</td><td></td><td>2.28</td></tr><tr><td>Platform</td><td></td><td>7.70</td></tr><tr><td colspan="3">Distances</td></tr><tr><td colspan="3">MPM stroke = d<sub>x</sub> = d<sub>y</sub> = 4 mm</td></tr><tr><td>d<sub>1</sub> = 11.9 mm</td><td>d<sub>2</sub> = 17.3 mm</td><td></td></tr><tr><td>d<sub>3</sub> = 44 mm</td><td>d<sub>4</sub> = 170 µm</td><td></td></tr><tr><td>d<sub>5</sub> = 305 µm</td><td>d<sub>6</sub> = 200 µm</td><td></td></tr><tr><td>d<sub>7</sub> = 237 µm</td><td>d<sub>8</sub> = 3 mm</td><td></td></tr><tr><td>d<sub>9</sub> = 1.2 mm</td><td>d<sub>10</sub> = 4 mm</td><td></td></tr><tr><td colspan="3">d<sub>11</sub> = 1.4 mm</td></tr></tbody></table>

## 2.2. Working Principle of the Conveyance System

The mobile part of the proposed conveyance system is composed of a platform of size 84 mm × 84 mm that is moved by the actuator array (Figure 3). The bottom side of the platform is structured with an 11 × 11 grid of stud patterns, and each pattern is composed of four square studs (visible in the bottom view of Figure 3a). On the top face of each MPM, a cylindrical stud is fixed and used to push the studs of the mobile platform. To facilitate the displacement of the platform and ensure its discrete positioning, balls are placed on the top side of the fixed part of the device. The balls have a 3 mm diameter and are placed in 2 mm deep holes located between the elementary actuators (Figure 3b). The platform is placed on the balls, and the distance between the top surface of the stud on the MPMs and the bottom side of the platform (d<sub>7</sub>) is theoretically at least 237 µm (Figure 3a). Owing to the balls and the platform geometry, the mobile platform can only reach discrete positions every 4 mm in the xy-plane. To achieve this 4 mm indexed displacement, several geometric design constraints were considered. In particular, the stroke of each actuator (d<sub>x</sub> and d<sub>y</sub> in Figure 2) was fixed at 4 mm, the distance between adjacent cavities (d<sub>3</sub>) was set to a multiple of 4 mm, and the periodicity of the platform patterns was defined as 2 mm × 4 mm along both x and y directions to ensure periodic indexing. A total of 14 × 14 discrete positions can be reached by the platform, which correspond to a reachable workspace of 56 mm × 56 mm in the xy-plane.

When it is placed in a given discrete position, the center of each platform pattern, composed of four square studs, is placed on the vertical axis of the corresponding ball due to gravity. Each ball can roll inside its cavity to minimize the force required to move the platform. During its movement, the platform undergoes a small vertical displacement along the z-axis. This vertical displacement t<sub>z</sub> is inherent to the mechanical indexing principle, which relies on the indexing stud of the platform sliding on the indexing. During indexing, this interaction results in a constant vertical translation of approximately 138 µm, which is independent of the payload. This vertical displacement t<sub>z</sub> can be analytically derived using the following geometrical relationship:

$$t_z = \frac{\sqrt{d_8^2 - d_9^2} - \sqrt{d_8^2 - 2d_9^2}}{2} \quad (1)$$
While increased payload may affect the dynamics of motion and the force required to initiate displacement, it also increases the indexing force and does not alter the final indexed position.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Schematic views of the conveyance system: (a) top view and (b) side view (C-C).</figcaption></figure>

Several holes have been made in the mobile platform structure in order to minimize its weight and therefore the force needed to move it. In addition, the mobile platform dimensions have been chosen so that it covers 2 × 2 elementary actuators, which correspond to the minimum number of actuators needed to generate the platform motion. The platform displacement along the x-axis is obtained using the displacement sequence shown in Figure 4 and the control signals given in Table 2. In the first step (Figure 4a), the two MPMs on the right side (MPM₁ and MPM₂) are switched along the +x-axis so that their studs push against the studs of the mobile platform, which moves to the right to the next discrete position of the platform (black arrows). The current used to move these two MPMs along the x-axis is called “pushing current” (Iₚ). At the same time (first step), the two MPMs on the left side (MPM₃ and MPM₄) are switched along the -x-axis in order to place them in the adequate position for the next step (red arrows). The current used to move these MPMs is called “moving current” (Iₘ). During MPMs switching (both for moving the platform—black arrows—and for MPM positioning—red arrows), a current can be used to generate a force orthogonal to the displacement axis (i.e., along the y-axis) to ensure that the MPM maintains lateral contact with the cavity side. This current is called “holding current” (Iₕ).
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Schematic representation of the different steps of the conveyance sequence and corresponding control signals: (a) two actuators on the right pushing the platform, (b) actuators repositioning themselves, (c) two actuators on the left pushing the platform, and (d) actuators repositioning themselves.</figcaption></figure>

<table><caption>**Table 2.** Control signals for a displacement sequence.</caption><thead><tr><th rowspan="2"></th><th colspan="2">MPM<sub>1</sub></th><th colspan="2">MPM<sub>2</sub></th><th colspan="2">MPM<sub>3</sub></th><th colspan="2">MPM<sub>4</sub></th></tr><tr><th>I<sub>x</sub></th><th>I<sub>y</sub></th><th>I<sub>x</sub></th><th>I<sub>y</sub></th><th>I<sub>x</sub></th><th>I<sub>y</sub></th><th>I<sub>x</sub></th><th>I<sub>y</sub></th></tr></thead><tbody><tr><td>I (a)</td><td>-I<sub>m</sub></td><td>-I<sub>h</sub></td><td>+I<sub>p</sub></td><td>+I<sub>h</sub></td><td>-I<sub>m</sub></td><td>+I<sub>h</sub></td><td>+I<sub>p</sub></td><td>-I<sub>h</sub></td></tr><tr><td>II (b)</td><td>-I<sub>h</sub></td><td>+I<sub>m</sub></td><td>+I<sub>h</sub></td><td>-I<sub>m</sub></td><td>-I<sub>h</sub></td><td>-I<sub>m</sub></td><td>+I<sub>h</sub></td><td>+I<sub>m</sub></td></tr><tr><td>III (c)</td><td>+I<sub>p</sub></td><td>+I<sub>h</sub></td><td>-I<sub>m</sub></td><td>-I<sub>h</sub></td><td>+I<sub>p</sub></td><td>-I<sub>h</sub></td><td>-I<sub>m</sub></td><td>+I<sub>h</sub></td></tr><tr><td>IV (d)</td><td>+I<sub>h</sub></td><td>-I<sub>m</sub></td><td>-I<sub>h</sub></td><td>+I<sub>m</sub></td><td>+I<sub>h</sub></td><td>+I<sub>m</sub></td><td>-I<sub>h</sub></td><td>-I<sub>m</sub></td></tr></tbody></table>

I<sub>h</sub>: Holding current; I<sub>m</sub>: moving current; I<sub>p</sub>: pushing current.

At the second step (Figure 4b), all the MPMs are switched along the y-axis so that the studs of the two MPMs on the right side are disengaged from those of the mobile platform and the studs of the two MPMs on the left side are engaged with those of the platform. The current used in this step is called a “moving current” (I<sub>m</sub>).

At the third step (Figure 4c), the two MPMs on the left side are controlled using a “pushing current” (I<sub>p</sub>) to move the platform to the right, to the next discrete position. At the same time, the two MPMs on the right side are controlled using a “moving current” (I<sub>m</sub>) to the left, to be in an adequate position for the next step.

At the fourth step (Figure 4d), the MPMs are moved along the y-axis to disengage the studs of the two MPMs on the left and to engage the studs of the two MPMs on the right. At the end of the step, all the MPMs are back in their positions, as shown in Figure 4a, and a new displacement sequence can be executed.

In the present article, an actuator array composed of 3 × 3 digital actuators was considered to demonstrate the ability to transfer the mobile platform from one pair of actuators to another in both directions. Since the mobile platform is moved by 2 × 2 actuators at a time, a 3 × 3 array is the minimum configuration that enables such transitions. Moreover, the control of the system is based on elementary current pulses that enable
motion of the mobile platform in both directions. To achieve this, several actuation sequences have been defined based on elementary pulses applied in a given order to achieve a displacement step in one direction or the other or to reposition the actuators prior to performing a new step. This pulsed-mode control also represents an originality of the proposed micro-conveyance device.

### 3. Actuator Modeling

A dynamic model capable of determining the MPMs and platform displacements has been developed to design the actuator, considering the magnetic force ($F_m$), electromagnetic force ($F_e$), friction force ($F_f$), and indexing force ($F_i$). The magnetic and electromagnetic forces calculations are based on previous works presented in [22,27] and are computed using Equations (2)–(4) where $\sigma_m$ is the surface charge density, $B_{ext}$ is the magnetic flux density from FPMs (for $F_m$) or from the MPM (for $F_e$), S is the surface of the MPM poles, and I is the current. Equation (3) gives the expression of the magnetic force exerted between two PMs. However, because in most cases, it is not possible to evaluate (2) analytically, the magnetic force is obtained by discretizing the surfaces S of the considered PM in p surfaces $\Delta A_p$. The three components of the magnetic flux density ($B_{ext}$) generated by a parallelepiped-shaped PM is computed using (5) where ($x_2-x_1, y_2-y_1, z_2-z_1$) corresponds to the PM dimensions. To compute these forces, the PM shapes are considered as perfect (parallelepiped shape) and the magnetization perfectly uniform and orientated along the vertical axis. The cross-section of the electrical conductors is not considered in the model; the conductors are simply considered as lines. With 1 A current, the electromagnetic force ($F_e$) value is 13.5 mN along the x-axis and 12.8 mN along the y-axis. The difference between these two forces is due to the vertical distance (along the z-axis) between the two sets of wires. Depending on the relative positioning of the MPMs within their cavities, an exhaustive numerical analysis over all possible configurations of the MPMs predicts that the magnetic holding force ranges from 1.23 mN to 9.52 mN, with an average value of 8.31 mN computed over the full configuration set. This is sufficient to ensure stable positioning of all the MPMs. The methodology used to design the balancing magnets configuration is detailed in previous work [26]. The relative positioning between the FPMs and the MPM along the z-axis (using wedges as visible in Figure 2c) ensures a vertical magnetic force ($F_{m_z}$) of 12.73 mN, which partially compensates the weight of the MPM and reduces the friction force (more precisely, the horizontal friction force $F_{hf}$, as described in the next paragraph). The three components of the magnetic force as a function of the MPM position along the x-axis are shown in Figure 5. In this figure, the two discrete positions are located at -2 and +2 mm. These discrete positions correspond to the extreme values (minimum and maximum) of $d_x$ reported in Figure 3. The x-axis component is almost linear as a function of the x-axis position and is negative for positions below zero and positive for positions above zero (Figure 5a). The y and z-axis components are not constant as a function of the x-axis position. The amplitude of these components is maximal for the discrete positions and minimal in the middle between the discrete positions (0 mm).

$$F_m = \oint_S \sigma_m \mathbf{B}_{ext} ds \qquad (2)$$

$$F_m = \sum_p \sigma_m(x_p) \mathbf{B}_{ext}(x_p) \Delta A_p \qquad (3)$$

$$F_e = I \int_{wire} d\mathbf{l} \times \mathbf{B}_{ext} \qquad (4)$$
$$B_x(x, y, z) = \frac{\mu_0 M}{4\pi} \sum_{k=1}^{2} \sum_{m=1}^{2} (-1)^{k+m} \ln(F(x, y, z, x_m, y_1, y_2, z_k))$$

$$B_y(x, y, z) = \frac{\mu_0 M}{4\pi} \sum_{k=1}^{2} \sum_{m=1}^{2} (-1)^{k+m} \ln(H(x, y, z, x_1, y_2, y_m, z_k))$$

$$B_z(x, y, z) = \frac{\mu_0 M}{4\pi} \sum_{k=1}^{2} \sum_{n=1}^{2} \sum_{m=1}^{2} (-1)^{k+m+n} \times \tan^{-1} \left( \frac{(x-x_n)(y-y_m)}{(z-z_k)} g(x, y, z; x_n, y_m, z_k) \right) \quad (5)$$

$$F(x, y, z, x_m, y_1, y_2, z_k) = \frac{(y-y_1) + [(x-x_m)^2 + (y-y_1)^2 + (z-z_k)^2]^{1/2}}{(y-y_2) + [(x-x_m)^2 + (y-y_2)^2 + (z-z_k)^2]^{1/2}}$$

$$H(x, y, z, x_1, y_2, z_k) = \frac{(x-x_1) + [(x-x_1)^2 + (y-y_m)^2 + (z-z_k)^2]^{1/2}}{(x-x_2) + [(x-x_2)^2 + (y-y_m)^2 + (z-z_k)^2]^{1/2}}$$

$$g(x, y, z; x_n, y_m, z_k) = \frac{1}{[(x-x_n)^2 + (y-y_m)^2 + (z-z_k)^2]^{1/2}}$$

<figure><img src="image_8.png" /><figcaption>Figure 5. (a) Evolution of the three components of the magnetic force (Fx, Fy, Fz) exerted on the MPM for different positions (see $d_x$ on Figure 3) between the two discrete positions (located at -2 and +2 mm) and (b) representation of the rotation angle $\theta$ and amplitude of the vertical indexing motion $t_z$.</figcaption></figure>

The friction forces generated by the mechanical contacts between the MPM and the fixed cavity are considered with a Coulomb model. Friction is assumed to be uniform, with identical contact conditions across all the balls, and an even distribution of the plate weight among them. Two components of the friction forces have been considered: the friction force ($F_{hf}$) due to the horizontal contact between the bottom side of the MPM and the glass layer and the friction force ($F_{vf}$) due to the vertical contact between the lateral side of the MPM and the cavity wall. Equations (6) and (7) give the expressions for these two friction forces considering a MPM displacement along the x-axis. $F_{vf_x}$ and $F_{hf_x}$ represent the friction forces along the x-axis, $\mu_v$ and $\mu_h$ the friction coefficients, $W_{MPM_z}$ the weight of the MPM, $F_{m_z}$ the magnetic force component along the z-axis, and $F_{e_z}$ the electromagnetic force component along the z-axis. The friction coefficients were measured experimentally using
an inclined plane method with a platform mounted on a rotary stage. The materials under consideration (a glass slide or a PMMA piece) were placed on this platform and gradually tilted using the stage. The parts were in direct contact with each other without lubrication. The maximum angle before slipping was recorded, and each measurement was repeated 12 times. Based on this experimental setup, the friction coefficients were found to be equal to $\mu_h = 0.41 \pm 0.07$ (for the horizontal contact between MPM (nickel coating) and glass) and $\mu_v = 0.27 \pm 0.06$ (for the vertical contact between MPM (nickel coating) and PMMA). The indexing force of the mobile platform obtained through the interaction between the studs and the balls is calculated with Equation (8).

$$F_{hf_x} = \mu_h \times (W_{MPM_z} - F_{m_z} - F_{e_z}) \qquad (6)$$

$$F_{vf_x} = \mu_v \times F_{my} \qquad (7)$$

$$F_i = \frac{W_p \times (\sin(\theta) + \mu_b \cos(\theta))}{\cos(\theta) - \mu_b \sin(\theta)} \qquad (8)$$

with $W_p$ the weight of the mobile platform, $\theta$ the rotation angle (defined in Figure 5b), and $\mu_b$ the friction coefficient between the ball and the dots ($\mu_b = 0.015$), respectively. It is assumed that the balls roll without slipping in the holes in which they are placed. The rotation angle $\theta$ depends on the system geometry such as the ball diameter and the studs' dimensions. When the mobile platform is in a discrete position, the indexing force is 53.5 mN. To initiate the MPM motion, the force to overcome is the sum of the magnetic holding force, the horizontal friction force, the vertical friction force, and the indexing force. This total force has been estimated to be 46.5 mN (considering an average holding force of 8.31 mN), of which the indexing force accounts for 26.8 mN. Because the magnetic force is not constant (minimum = 1.23 mN and maximum = 9.52 mN), the total force to be overcome varies from 37.5 mN to 48.0 mN. Using the electromagnetic force conversion factor of 13.5 mN for 1 A, the minimum current required to move the platform is therefore estimated to be 3.44 A (minimum = 2.77 A and maximum = 3.55 A).

Considering a displacement along the x-axis, the total force exerted by each MPM can be determined using Equation (9) and then the mobile platform displacement can be calculated using Equation (10), where $m_p$ is the mass of the platform. To compute the platform displacement, it is assumed that two MPMs (named MPM1 and MPM2 in (10)) push the platform at the same time. The rebound phenomenon that occurs when the magnet impacts the edges of the cavity is not taken into account in the model. This model was implemented in Matlab 2020 and was used to size the system and a comparison between simulated and experimental results is provided in the next section. According to the IPC-2221 standard, it was determined that, for a continuous current of 5 A, the temperature rise of the conductor would be 55 °C. Since our driving mode is not continuous but pulsed, the temperature rise is expected to remain well below 55 °C, implying that the MPM temperature remains below its maximum allowable temperature before demagnetization (80 °C). This is further ensured by the fact that the conductors are not in direct contact with the MPM, as a glass layer is placed between them.

$$F_{MPM_x} = F_{e_x} - F_{m_x} - F_{hf_x} - F_{vf_x} \qquad (9)$$

$$d_{p_x} = \frac{1}{m_p} \iint (F_{MPM1_x} + F_{MPM2_x} - F_{i_x}) dtdt \qquad (10)$$
# 4. Experimental Setup and Results

## 4.1. *Experimental Setup*

To experimentally validate the proposed concept, a prototype was manufactured (Figure 6). The mechanical structure of the device is composed of several stacked PMMA plates with cavities for the PCBs, MPMs, and FPMs, balancing magnets and balls. These cavities were obtained using a laser cutting technique. The control of the device is implemented with a desktop computer equipped with two input/output boards (National Instruments PCI 6289 and PCI 6733) and with an interface developed in Labview 2012. With these I/O boards, four voltage signals are generated and converted into current signals using four voltage-to-current converters. (input: ±10 V; output: ±7 A; bandwidth: 50 kHz). To limit the number of converters and control signals, a control module presented in [28] was used. This module enables the selection of the actuators to be controlled via digital signals (generated by the I/O boards), and the current signals are then injected into the selected actuators. Two current signals are used to switch the actuators along the x-axis and two others along the y-axis. Moreover, a high-resolution camera (Point Grey Grasshopper 3) is positioned above the system to observe and measure the displacement of the MPMs and the mobile platform. The camera was calibrated using a grid calibration target from Edmund Optics (Ref. #62-949), composed of circular markers with a diameter of 62.5 µm and a spacing of 125 µm. All the measurements provided in the next sections were captured using this camera. The platform motion was measured by tracking five markers arranged in a cross pattern (see Figure 6) and using image processing algorithms.

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Prototype of the conveyance system.</figcaption></figure>

## 4.2. *Characterization of 1D Displacement*

The proposed actuation concept was first validated for 1D displacement along both x and y axes. In Figure 7, three full displacement sequences of the mobile platform along the x-axis and four images of the platform position are shown. These three sequences
lead to six displacement steps. For this test, a 5 A pushing current ($I_p$) is selected slightly above the minimum actuation threshold (~4 A) and below the power supply limit (7 A). A pulse duration of 100 ms is used, providing sufficient margin with respect to the ~50 ms switching time required for the MPM to reach its final position. For the moving current ($I_m$), 4 A is applied since the platform is no longer actively pushed, while a 1 A holding current ($I_h$) is used to maintain the MPM in contact with the lateral side of the cavity.

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Displacement of the mobile platform along the x-axis: (a) displacement as a function of time and (b) pictures showing the platform displacement.</figcaption></figure>

For each sequence, the four sub-steps, named “I”, “II”, “III”, and “IV” (as represented in Figure 4), are executed successively and waiting times are also added between each motion step. During the sub-steps named “I” and “III”, two MPMs push the platform to its next position. During these steps, the platform is moved. For the sub-steps named “II” and “IV”, the four MPMs position themselves for the next steps; they do not push the platform. As shown in this figure, it can be observed that the proposed discrete conveyance strategy has been validated experimentally and that the device is able to generate several displacement steps.
### 4.3. Characterization of 2D Displacement

The ability of the device to generate planar motions is shown in Figure 8, where each dot corresponds to a platform position measurement. In this figure, a square trajectory was performed with two sub-steps in both directions and with the same control signals as for 1D displacement. This counterclockwise square trajectory was repeated three times. For this experiment, each pixel of the image represents an area of 5 × 5 µm². The measured trajectories present some errors, especially during the transition between two discrete positions, and the maximum straightness error was measured to be 119 µm for displacement along the x-axis and 102 µm along the y-axis. The straightness errors are mainly due to inhomogeneity in the pushing force between the actuators and the geometric manufacturing tolerances, which induce small rotations and deviations of the platform during its displacement. These rotations have not been directly measured; nevertheless, this could be considered to characterize more deeply the transition phase between two discrete positions in order to identify precisely the origin of the straightness error and to propose solutions (either in the design or in the control of the device) to reduce it. However, one of the main advantages of the proposed conveyance system remains the positioning repeatability at each discrete position, ensured by the indexing provided by the balls. For this reason, the positioning repeatability error was characterized and will be presented in Section 4.5.

<figure><img src="image_11.png" /><figcaption>**Figure 8.** The 2D displacement of the mobile platform (arrows indicate the direction of rotation).</figcaption></figure>

### 4.4. Comparison Between Experimental and Simulated Results

In this section, the displacement of the mobile platform between two discrete positions has been simulated and measured experimentally as a function of time. A comparison between simulation and experimentation is presented in Figure 9 for three pushing current values (5, 6, and 7 A). It can be observed that the switching time between two discrete positions is reduced when the pushing current increases. The reason is that a higher current induces an increase in the applied electromagnetic force and thus the MPM acceleration, thereby reducing the switching time. For the three current values, a good correspondence can be observed between simulated and experimental results. A detailed analysis of the actuators' dynamic behavior is also provided in previous work [22].
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Comparison between experimental and simulated mobile platform displacements (with 5 A, 6 A, and 7 A pushing current values).</figcaption></figure>

4.5. Characterization of the Conveyance System

In this section, some performance metrics of the conveyance system are characterized and discussed. The first parameter is the positioning repeatability error of the mobile platform at discrete positions. For this, 30 back and forth displacements were performed along the x and y axes, and the positioning repeatability errors have been determined. The positioning repeatability error is defined as the standard deviation of the x or y positions of the platform in the discrete positions. For this test, the camera field of view has been reduced, and each pixel of the image represents an area of 2.5 × 2.5 µm². The same control signals were used as in paragraphs IV.B and IV.C (I<sub>p</sub> = 5 A, I<sub>m</sub> = 4 A, I<sub>h</sub> = 1 A, pulse duration = 100 ms). Figure 10a,b represent the 30 reached positions of the platform for a displacement along the x and y-axis, respectively. Based on this figure, the positioning repeatability errors have been determined and are indicated in Table 2. It can be observed that repeatability is consistent for the two displacement directions and for both axes. The average repeatability error is 5.5 µm, which shows the efficiency of the proposed indexing method compared to the displacement step of 4 mm.

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Positioning repeatability of the mobile platform: (a) displacement along the x-axis and (b) displacement along the y-axis.</figcaption></figure>

Table 3 provides a review of the conveyance system performances. The maximum conveyance velocity has been experimentally determined and found to be of the order of 16 mm.s⁻¹ for both axes. This velocity has been obtained with the maximum current amplitude (7 A). The conveyance velocity is higher for the x-axis than for the y-axis because
of the force difference between the two displacement axes (as mentioned in Section 2). Another parameter that was characterized is the maximum transportable mass of the conveyance system as a function of the pushing current amplitude (Figure 11). A set of current amplitudes (12 distinct values, as indicated in Figure 11) was tested, and for each amplitude, non-ferromagnetic masses were placed on the platform with 0.1 g increments. The platform motion was evaluated by performing the displacement sequence at nine distinct platform locations, each corresponding to the platform being positioned above a 2 × 2 set of elementary actuators (four actuators). For each payload mass and pushing current amplitude, one trial was performed at each location (nine trials per mass-current condition). A trial was deemed a success when the platform completed the sequence and reached the next discrete indexed position; otherwise, it was recorded as a failure. The success rate for a given mass-current condition was then defined as the number of successful locations out of nine (0–9, i.e., 0–100%). Based on this outcome, three operating zones were defined: Zone 1 corresponds to 9/9 successes (100%), Zone 3 to 0/9 successes (0%), and Zone 2 to intermediate results (1–8/9). The partial success rates observed in Zone 2 are attributed to location-dependent variations among the elementary actuators (e.g., dimensional tolerances, magnetization variability, and local friction conditions). From this test, a maximal transportable mass of 15 g was determined for a 7 A current. In this figure, the minimum current required to initiate platform motion is observed to be approximately 3.5 A. This value is consistent with the simulated value reported in the modeling section.

<table><caption>**Table 3.** Performances of the conveyance system.</caption><thead><tr><th colspan="3">Positioning Repeatability Error</th></tr><tr><th></th><th>Displacement along the x-axis</th><th>Displacement along the y-axis</th></tr></thead><tbody><tr><td>Repeatability error along the x-axis</td><td>4.4 µm</td><td>6.4 µm</td></tr><tr><td>Repeatability error along the y-axis</td><td>4.0 µm</td><td>7.1 µm</td></tr><tr><td colspan="3">Straightness error</td></tr><tr><td>Along the x-axis</td><td colspan="2">118.7 µm</td></tr><tr><td>Along the y-axis</td><td colspan="2">102.2 µm</td></tr><tr><td colspan="3">Conveyance velocity</td></tr><tr><td>Velocity along the x-axis</td><td colspan="2">Up to 16.4 mm.s<sup>-1</sup></td></tr><tr><td>Velocity along the y-axis</td><td colspan="2">Up to 15.9 mm.s<sup>-1</sup></td></tr><tr><td colspan="3">Transportable mass</td></tr><tr><td>Maximum transportable mass</td><td colspan="2">Up to 15 g</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Transportable mass as a function of the pushing current amplitude.</figcaption></figure>
## 5. Comparison with Conveyance Systems from the Literature

Several criteria must be considered for a micro-conveyor system depending on the configuration, architecture, machining stations, objects to be conveyed, and desired performances of the micro-factory where it is integrated. The number of degrees of freedom, the scalability to cover large surfaces, and performance aspects such as speed and positioning accuracy must be carefully considered. Moreover, the integration of a micro-conveyor should be simple in order to minimize the overall complexity of the micro-factory. The actuation principle (analog or digital) therefore represents a crucial parameter that influences the control complexity, the need for sensor integration, and the energy consumption, as mentioned in Section 1. To highlight the interest of the proposed device, Table 4 provides a comparison with other solutions reported in the literature, considering both performance indicators and ease of integration. In terms of performance, our system can convey a mass up to 15 g with intermediate conveyance speeds and relatively low errors compared to other systems. Given the general requirements and performances of micro-conveyors, the proposed system stands as a viable solution within the field of micro-conveyance.

<table><caption>**Table 4.** Characteristics and performances comparison with planar actuators from the literature.</caption><thead><tr><th>Ref</th><th>Scalable</th><th>DoF</th><th>Physical Principle</th><th>Actuation Principle</th><th>Conveyed Object</th><th>Speed (mm.s<sup>-1</sup>)</th><th>Positioning Resolution and Other Errors</th></tr></thead><tbody><tr><td>[7]</td><td>Yes</td><td>2</td><td>Electrothermal</td><td>Digital</td><td>7 mm × 5 mm</td><td>0.67</td><td>Pos. Res. ~2 µm</td></tr><tr><td>[12]</td><td>Yes</td><td>3</td><td>Piezoelectric</td><td>Digital</td><td>500 µm × 500 µm</td><td>20</td><td>Pos. Res. ~300 µm</td></tr><tr><td>[13]</td><td>Yes</td><td>2</td><td>Electrostatic</td><td>Analog</td><td>132 mm × 132 mm, 11 g</td><td>250</td><td>Pos. Res. &lt;400 µm</td></tr><tr><td>[16]</td><td>Yes</td><td>3</td><td>Pneumatic</td><td>Analog</td><td>Ø5 mm, 19.43 mg</td><td>137</td><td>Pos. Rep. 17.7 µm<br>Pos. Res. 0.3 µm</td></tr><tr><td>[17]</td><td>Yes</td><td>3</td><td>Pneumatic</td><td>Analog</td><td>Ø150 mm</td><td>300</td><td>Pos. Error ~248 µm</td></tr><tr><td>[22]</td><td>Yes</td><td>3</td><td>Electromagnetic</td><td>Digital</td><td>20 mm × 20 mm, 0.21 g</td><td>2</td><td>Pos. Rep. 4.7 µm<br>Straig. 5.2 µm</td></tr><tr><td>[29]</td><td>No</td><td>3</td><td>Electromagnetic</td><td>Analog</td><td>78 mm × 78 mm</td><td>0.1</td><td>Pos. Res. ±1 µm</td></tr><tr><td>Present work</td><td>Yes</td><td>2</td><td>Electromagnetic</td><td>Digital</td><td>84 mm × 84 mm, up to 15 g</td><td>16</td><td>Pos. Rep. 6 µm<br>Straig. 110 µm</td></tr></tbody></table>

## 6. Conclusions and Perspectives

In this paper, a planar conveyance device based on a 3 × 3 digital electromagnetic actuator array is presented. The mobile platform of the device can reach discrete positions distributed in the xy-plane and is moved by means of a collaborative actuation principle between the digital actuators. The conveyance of the mobile platform is generated with four elementary actuators and is achieved through a displacement sequence composed of two sub-steps. The design of the system was carried out using a dynamic model that considers the magnetic, electromagnetic, and mechanical effects and predicts the kinematic behavior of the mobile platform. A prototype was fabricated and characterized. The proposed principle was validated experimentally for 1D and 2D motions, and the performances were characterized in terms of the straightness error, positioning repeatability, conveyance velocity, and transportable mass. A comparison between the theoretical and experimental results was also carried out, showing similar behavior.

In the future, a miniaturized version of the proposed conveyance system will be designed and manufactured to minimize the displacement step of the mobile platform. In addition, future work will strengthen the actual quantitative model-experiment validation, providing a deeper understanding of the system and facilitating future design sizing.
Moreover, the indexing solution with the balls induces a vertical displacement of the mobile platform during indexing. This vertical displacement may disturb the functioning, and a new version that avoids this displacement will be investigated. The proposed concept provides only two degrees of freedom (translations along the x and y axes) of the mobile platform and a solution to provide three degrees of freedom (adding a rotation along the z-axis) is under investigation to improve the functionality of the proposed concept.

**Author Contributions:** Conceptualization, E.D., L.P. and C.P.; methodology, E.D. and L.P.; software, C.B. and G.G.; validation, C.B. and G.G.; formal analysis, C.B. and G.G.; investigation, C.B. and G.G.; resources, C.B., G.G. and N.P.; data curation, C.B. and G.G.; writing—original draft preparation, C.B., G.G., E.D., L.P., N.P. and C.P.; writing—review and editing, E.D., L.P. and C.P.; visualization, E.D., L.P. and C.P.; supervision, E.D., L.P. and C.P.; project administration, L.P.; funding acquisition, E.D., L.P. and C.P.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

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