A Low-Cost Magnetic 2D Tracking System for Mobile Devices as an Alternative to Large Interactive Tabletops

Abstract

In the last decade, interactive tabletops have emerged as a hardware solution for collaborative interactions, providing shared workspaces that support group learning and work. However, despite a variety of studies highlighting their benefits, adoption in educational and professional environments remains limited due to high cost, weight, and spatial constraints. This paper presents an alternative hybrid approach of augmenting static surfaces (e.g., printed images or plans) with off-the-shelf mobile devices through a dynamic peephole interaction. The system uses a rotating, asymmetric static magnet and magnetometers commonly found in all mobile devices, requiring solely software on the device to calculate its relative position based on field strength and relative angle. This first prototype is affordable (∼€10), easy to build with a minimal set of components (e.g., LEGO or 3D-printed parts), device-independent, and offers an accuracy of 1.4 cm, with potential for improvements both to accuracy and the currently limited operating range of 30 cm.

1. Introduction

Collaboration has been hailed as a 21st-century skill by both industry and education [1,2]. The design of software and hardware to support collaboration (and collaborative learning) has a long history. Since the 1980s, groupware has emerged as a term designating systems that support group work [3]. In the 1990s, interactive tabletops attracted the interest of researchers as devices to support collaboration due to the shared workspace that allows multiple users to interact simultaneously. While initially developed entirely by researchers as prototypes [4], in the 2010s, large touchscreens (40” or bigger) became commercially available (e.g., DiamondTouch [5], Samsung Flip [6]). However, to this day, such devices are rarely found in either classrooms or companies, and are often restrained to settings like shopping centers, museums, or showrooms [7]. This is also evidenced by the fact that no publicly available data exist quantifying their actual adoption or penetration rates in classrooms, unlike more common technologies such as interactive whiteboards or mobile devices [8]. Some of the key challenges for adoption are the relatively high investment cost (€2000 and above), substantial space and weight, the need for a power supply, and the limited availability of suitable software for this hardware type (e.g., multi-user and multi-touch support, etc.). Together, this limitations have hindered widespread adoption. Relatedly, the low device adoption rate exacerbates the problem of software availability. Other niche device types that need special adaptation of software share this problem: if the adoption rate of these devices stays negligible, developers are likely to focus on largely adopted device types instead. This challenge is evidenced by the failed adoption of devices such as the Windows Phone, Google Glass, Razer Hydra, or Microsoft Kinect, which (among other reasons) faced limited software development due to their low adoption rates, leading developers to prioritize more widely used platforms instead [9,10]. In the case of interactive tabletops for educational settings, the rise in alternative teaching and learning models (e.g., situated and mobile learning [11,12]) has further reduced the contexts in which large interactive touchscreens can be effectively employed, shrinking their practical usage time and thus weakening the case for investment despite their potential for collaborative learning [13,14].

Concurrently, during the last two decades, pedagogic innovation has shifted in focus to off-the-shelf technology such as smartphones and tablets in both traditional and situated learning [15], with different approaches to integrate mobile devices into a shared collaborative space. Collaborative spaces can, for instance, be established by using a single shared display, partitioning the available screen space into individual areas for each user. Alternatively, a combination of multiple devices can be used to create a common collaborative space. In both configurations, available space can be either partitioned beforehand or be presented to the group without dedicated ownership. Since the 1990s, this idea has been formalized in the Single Display Groupware (SDG) model, which provides a shared interface for co-located users [16]. However, this approach has been explored mostly on personal computers and large projection screens [17,18], with far fewer examples of its application on mobile devices. One such example, presented by Čarapina and Pap [19], demonstrates the use of a split-screen approach for mathematics assignments in primary education. The authors demonstrated how multiple co-located students could simultaneously collaborate around a single tablet.

Still, only a limited number of studies have addressed the challenges of complex content interaction that arise from smaller display sizes, compared to tabletops, and the shared use of the working surface. Simon et al. [20], for instance, observed that in activities with mobile devices, group members would pick up the device, previously in the role of a common support, excluding other members from viewing and interacting with the device. Similar findings have been reported in several other studies, underscoring the importance of interaction design in supporting shared workspace contexts across different educational levels, from primary [21] to secondary [22] and higher education, and collaborative work settings.

Building on these concerns about how small devices mediate shared interaction, another line of work explored how interaction techniques can expand or reconfigure the effective workspace of handheld displays. A notable example is the system designed by Spindler et al. [23], which explores spatial interaction with handheld displays tracked in three-dimensional space using a consumer depth camera (i.e., Microsoft Kinect). The system builds on the concept of tangible magic lenses, allowing users to manipulate and explore complex information spaces by physically moving one or several handheld displays above a tabletop surface. This approach extends the interaction space beyond the two-dimensional plane and explicitly supports co-located parallel work and collaboration. Through spatial input, combined with surface and head input modalities, the system demonstrates how tangible displays can bridge digital and physical spaces, offering a low-cost and flexible alternative to complex setups. More recently, Babić et al. [24] investigated spatial interactions using off-the-shelf smartphones to enable distant display control without the need for specialized hardware or external tracking systems. Their approach demonstrates how everyday mobile devices equipped with built-in sensors, such as accelerometers, gyroscopes, and cameras, can be leveraged to support spatial awareness and mid-air gestures for interacting with large or remote displays. This work illustrates the potential of lightweight and scalable solutions for spatially aware interaction.

Classic, camera-based augmented reality (AR) technology is typically used for augmenting larger spaces, but exacerbates the issue of device appropriation, since it requires one user to hold the smartphone or tablet at a certain distance from the augmented surface, requiring other group members to gather around the device holder [25]. An alternative solution is for all group members to use their own devices to work in the augmented space [26], but this device-centric focus reduces awareness of others’ actions and diminishes socio-cognitive availability, a key component of effective collaboration [27]. Another issue with camera-based AR is the difficulty of interacting with virtual elements, as tablets have to be held with both hands, making touchscreen interaction difficult. This challenge also applies to smaller devices, such as smartphones, where free-hand interaction without physical support reduces precision and control. In both cases, muscle fatigue remains a problem over extended periods of usage [28].

To adress this issue, researchers such as Rädle et al. [29] have investigated the possibility of augmenting static surfaces, such as images, through mobile devices in direct, physical contact with the surface. Their system constitutes a spatially aware tabletop system that employs a depth-sensing camera mounted above a workspace to track the positions and orientations of multiple tablets or smartphones placed on the table. This setup enables ad hoc around-the-table collaboration, where users can move and align their personal mobile displays to form a larger shared workspace and seamlessly exchange digital content across devices. This type of interaction has later been termed SPART by Simon et al. for Surface-Positioning Augmented RealiTy [20]. Conceptually, this interaction design can be situated within the broader category of peephole interactions, which employ a small, interactive device as a movable window into a larger virtual space. Peephole interactions have been categorized by Mehra et al. [30] into dynamic and static peephole interactions (see Figure 1).

Static peephole interactions require users to perform an additional input action (e.g., panning and zooming on map applications such as Google maps (https://www.google.com/maps, accessed on 11 July 2025), whereas dynamic peephole interactions map rotation and/or position of the device to the virtual space, making it possible to investigate the virtual space through physical movements of the device itself (e.g., star gazing apps allowing users to identify star names and constellations, for example the Sky Map application (https://play.google.com/store/apps/details?id=com.google.android.stardroid, accessed on 11 June 2025). The latter interaction type allows for a spatial mapping of virtual and physical space [30]. SPART exploits this affordance to display virtual content on a physical medium (e.g., another virtual layer with dynamic information and interactive elements on a physical map). The relatively small size of the screen becomes a collaborative feature, turning the mobile device into a common group focal point through which all group members can view enhancements while maintaining awareness of each other’s actions and keeping their hands free (see Figure 2).

In this paper, we present SPART-MAR (on-Surface Augmented RealiTy–MAgnet Rotation), an accessible, robust, and mobile solution for 2D localization. We offer a new approach to implement the previously described SPART dynamic peephole interaction, where MAR (i.e., MAgnets in Rotation) denotes a simple mechanism in which a small rotating magnet produces a distinctive magnetic signal that can be reliably detected by magnetometers in mobile devices. By fitting this signal to an ellipse and detecting its orientation and polarity, SPART-MAR computes the device’s relative angle and distance to the magnet in real time. This spinning marker thus serves as a reference point, to which mobile devices can calculate their relative position. With this information, augmented content can be mapped to a static support (e.g., a printed image or map). SPART-MAR thus enables dynamic peephole interaction. This allows users to explore digital layers superimposed on printed supports. For example, in an educational collaborative learning scenario, a group of students can examine a printed geographic map together during a classroom activity, with the mobile device functioning as a movable viewpoint that enables dynamic exploration of the augmented surface as depicted in Figure 2. As learners move the device across the map, contextual digital layers, such as topographical data, population density, or historical overlays, appear on the device. At any given position, students can additionally engage with on-screen elements, including zooming into a selected area, accessing three-dimensional representations, or adding annotations. This combined interaction paradigm integrates spatial exploration and content manipulation, transforming the passive printed map into an interactive and collaborative learning artifact. While dynamic peephole interactions are common, SPART is particularly difficult to implement, due to its direct contact with the underlying surface, making it impossible to rely on classic alternatives, such as vision-based augmented reality. Other implementations, described in the following sections, have serious limitations, such as limited practicality due to the need for e.g., physical connections to the mobile device. An implementation with magnets thus offers wireless localization. Therein lies the novelty and contribution of our implementation and this paper.

The structure of the rest of this paper is as follows: in Section 2, we first present previous work and the existing SPART prototypes that informed the development of the new SPART-MAR prototype, alongside a set of criteria to be met to enact an accessible, robust, and mobile alternative to interactive tabletops with off-the-shelf mobile devices. Section 3 introduces the underlying principle of the SPART-MAR prototype and presents the circuit design and algorithm, while Section 4 describes the experimental setup used to test the prototype. Section 5 presents the results of this evaluation. Finally, Section 6 and Section 7 discuss limitations and outline avenues for future work.

2. Previous Work and SPART Prototypes

Localization techniques can be categorized into three different types: dead-reckoning, fingerprinting and geometric methods. Each of these abstract techniques has been implemented in the past through different technologies, exploiting different environmental properties (e.g., acoustic or electromagnetic waves, inertial movement of bodies, etc.). Consequently, a wide variety of prototypes and products exist for localizing objects accurately locally, each with its advantages and drawbacks. To systematically identify available technologies or products, Simon et al. [20] introduced the following requirements for SPART interactions in educational settings:

1.

Accuracy: To maintain the illusion of virtual elements in physical space, the mean Euclidean distance between real and calculated position should not exceed 1 cm.

2.

Fluidity: When moved across the underlying surface, an update rate of 10 Hz or better should be achieved.

3.

Minimum operating range: The A3 format is a standard size for printed media in Europe (29.7 × 42 cm). The technology should thus operate across such a surface.

4.

Mobility: To support either meetings or situated learning in a variety of settings, the technology should be compact and portable, with a weight of less than 500 g and dimensions not exceeding those of a notebook (A4).

5.

Affordability: For successful deployment in educational settings, the device cost should not exceed €50 per unit.

6.

Do-It-Yourself (DIY) & Reparability: The production and assembly of a prototype should be possible for students or educators. Components should be easily available.

7.

Multi-device support: Multiple devices can be located simultaneously (minimum two).

We leveraged these criteria to evaluate available solutions along the localization techniques of dead reckoning, fingerprinting and geometric methods.

Dead reckoning of mobile devices (using integrated measurement unit) relies on an initial, known position and history of position changes (e.g., in acceleration or relative rotation). However, the accumulating error in each change makes this approach inappropriate for fine-grained localization [31]. Similar techniques have been used with computer mice with similar issues [32].

Fingerprinting techniques have been used, for example, in indoor localization prototypes, relying on a calibration phase in which environment attributes (e.g., Wi-Fi signal strength) are assigned to positions in space (that then can be looked up in such a map by devices to access their current location by measuring attributes during the online phase). Technologies such as sonar and computer vision are frequently used for such approaches, but fail to enable low-cost solutions that are also accurate [33,34,35]. Simon et al. further evaluated technologies such as infrared codes and NFC tag grids that fail criteria of reparability, accuracy or fluidity [20].

Geometric techniques include a wide range of technologies. All are based on environmental properties that allow for extracting one of the following combinations: an angle and a length, two lengths or two angles relative to reference points in space. GPS is a typical implementation of such a technique, but is unsuitable due to its lack of accuracy in our context. Generally, wave-based implementations are sensitive to ambient noise and reflections. Beyond triangulation and trilateration, image recognition and depth sensing have been used in a wide variety of configurations. HuddleLamp is one such implementation, equipped with a depth-sensing camera to track the positions of multiple tablets on a table, enabling cross-device interaction [29]. Huddlelamp, however, requires an overarching lamp-like structure to track mobile devices beneath, with limited mobility.

Syed et al. [36] conducted a review on AR technologies, among which the authors listed tracking technologies, including magnetic, mechanical, acoustic, optical and inertial technologies.

Simon et al. [37] consequently introduced a family of prototypes (see Figure 3), based on their underlying technology: acoustic (SPART-AC), mechanical (SPART-ME), magnetic (SPART-MA) and optic (SPART-OP).

The class of acoustic prototypes (SPART-AC) relies on the difference-of-arrival of sound waves emitted by the mobile device with four microphones at known positions within the augmented surface (see Figure 3). The acoustic prototypes, however, can not, in their current state of development (11/2025), provide the accuracy with conventional tablets or smartphones that initial simulations predicted. Fluidity was restricted to position updates when the device was at rest, after a delay of around 3 s. Furthermore, the prototypes did not fulfill the requirements of mobility, robustness (ambient sound reducing accuracy), DIY (requiring specialized hardware) and affordability (>€300).

Recent mechanical versions (SPART-ME) provide sufficient fluidity, accuracy, operating range, mobility, robustness, and affordability, but the prototype is physically attached to the underlying surface and connected by a thread to the mobile device (see Figure 3). Although the components of this prototype are easily accessible or can be printed in 3D, the use of custom rotational sensors requires users to calibrate each of them in a lengthy procedure. Multi-device support is also limited, since one prototype is required for each mobile device, limiting the number of devices on one surface due to the number of concurrent threads spanning between prototypes and mobile devices.

In an attempt to avoid the restrictions of physical connections, Simon et al. [37] also prototyped a class of magnetic SPART prototypes (SPART-MA), using static Neodymium magnets as reference points for mobile device magnetometers. The prototype in this paper belongs to this class of SPART prototypes. While first tests with a single magnet and a fingerprinting technique showed promising results in terms of close-range accuracy, extending the surface with multiple magnets and mitigating the necessity to measure every position manually before using the system (offline phase) through the use of magnet models failed due to the heterogeneity of commercial magnets (varying field symmetry and strength). SPART-MAR intends to address both limitations of the mechanical versions (physical connection) and implement a magnetic version that is robust and does not require an extensive offline phase.

Finally, optical systems (SPART-OP) were considered, but not implemented due to restrictions such as cost (e.g., additional camera systems for tracking, exceeding the cost criterion), robustness or the DIY criterion.

3. SPART-MAR Prototype Design

In light of the limitations of prototypes found in literature and the SPART family (either relying on physical connections to the mobile device or suffering from environmental noise (SPART-AC), we investigated the possibility of a wireless solution using a magnetic signal (bypassing existing shortcomings from SPART-MA prototypes relying on static magnets).

3.1. Localization Technique and Technology

For the design of SPART-MAR, we build on a geometric localization technique relying on a single angle and distance (which enabled the compact design of SPART-ME). The challenge then can be centered around the type of information that can be used to obtain one angle and one distance relative to a reference point without a physical link. Ideally, any physical connection should be avoided while using integrated smartphone sensors to avoid external circuitry. We thus investigated the use of symmetric, rotating magnets. Rotating magnets produce a 2D signal in the sensor plane of magnetometers, forming an ellipse (see Figure 4).

The ellipse major runs through the north and south pole of the rotating magnetic field, therefore providing an angle. The length of the major is equivalent to the distance to the center of the field, decreasing in strength following an inverse square law. However, from the measured ellipse signal, it is unclear in which direction along the major axis of the ellipse the magnet is located. A symmetric field does not provide sufficient information on which direction the rotating magnet is located. Hence, we implemented the use of an asymmetric magnet, built from four bar magnets forming an overall arc (see Figure 5).

Arranging magnets in an arc leads to a slightly skewed magnetic field towards the center of the arc (see Figure 6). While the readings still translate into an ellipse, the skewed nature of the field influences the distance between measured data points, assuming a constant time resolution in sensor readings: When the field rotates at a constant speed clockwise, the transition from North to South Pole will take slightly longer than the transition from South to North. Thereby, the North and South poles can be distinguished, and one of the two possible vectors along the ellipse’s major axis can be excluded.

3.2. Circuit and Prototype Design

The prototype design consists of a geared DC motor. For the current setups, LEGO motors of type 43362 (datasheet at https://www.scribd.com/document/365143284/Lego-Motor, accessed on 15 May 2025) were chosen for their speed (60–120 rpm) and a consistent torque of 2.25 N · cm across its operating voltage). As Figure 7 illustrates, a potentiometer and power switch were added to the circuit for testing purposes. Attached to the down-pointing motor axle is a set of four N42 Neodymium bar magnets (datasheet available at https://www.supermagnete.de/data_sheet_Q-40-10-05-N.pdf, accessed on 4 May 2025) with a size of 10 × 40 × 5 mm each. Magnets are spaced out at one side for the set to form an arc of 45 degrees (see Figure 5).

The motor is housed in a simplistic LEGO frame to allow it to spin the magnets perpendicularly to the surface, at the same height as the mobile device (see Figure 8). The prototype is powered by two 1.5 V AA batteries. The casing can either be 3D printed or built with LEGO or other available materials that allow for positioning the magnet at a clearance of around 1 mm above the surface (see Figure 5).

3.3. Algorithm

To interpret the magnetic field data generated by SPART-MAR, a dedicated Android application was developed. The application continuously accesses the integrated magnetometer of the device, processes sensor readings (hard iron-calibrated), and transforms them into positional information relative to the spinning magnet. The algorithm for calculating the relative position of the device consists of three main stages:

1.

Calibration: Establish a function to map the ellipse major length to the distance.

2.

Ellipse detection: Establishes whether the device is in range of SPART-MAR.

3.

Ellipse fitting: Given the data of one rotation, determine the best-fitting ellipse’s major angle and length.

4.

Direction detection: Convert the ellipse major in a vector by determining in which of the two possible directions the prototype is located.

Steps 2 to 4 are performed with each sensor update. The following subsections detail each of these stages.

3.3.1. Calibration

Magnetic field strength decreases at a rate of $1/strengt{h}^2$ over distance. However, this does not strictly apply to asymmetric fields, and thus we generalize $f\left(m\right)=a·m^b$, where $f\left(m\right)$ is the distance in centimeters and m the measurement in $\mu T$, while a and b are the parameters to be found through calibration. The calibration procedure asks the user to put the device on a grid at distances from the rotating magnet’s axis that the user can define. After a minimum of three recorded couples of distance and major length recordings, a logarithmic regression is performed and the function parameters are deduced. The application allows multiple such configurations for different prototypes (and the different magnets used therein).

3.3.2. Ellipse Detection

The deployed algorithm uses an autocorrelation to identify oscillations in a buffer of 100 sensor (x) readings, cyclic patterns. For each lag r, the autocorrelation is calculated through:

$$R\left(\tau \right)={\displaystyle \frac{{\sum }_{t=0}^{N-\tau -1}(x_t-\overline{x})(x_{t+\tau }-\overline{x})}{{\sum }_{t=0}^{N-1}{(x_t-\overline{x})}^2}}$$

(1)

where $x_t$ represent the sensor values, $\overline{x}$ is the mean of the values, N is the number of readings and r is the lag. Each lag is then stored in a buffer, which is then scanned for local maxima. The first two maxima are used to determine the start and end of a full ellipse in the buffer data.

3.3.3. Ellipse Fitting

We initially tested the use of the most distant points in the ellipse for determining the major axis, but given the variety in sensor points (particularly for higher rotational speeds) and the fact that with increasing distance, even slight angular variations result in large variations of measured positions, we implemented the ellipse fitting algorithm by Halír, and Flusser [38]. The algorithm is stable for noisy data and non-iterative, with a capacity to fit over 100 k data points and yielding the major axis vector as output, making it an adequate choice for our context of limited resources (mobile context) and frequent updates. Possible alternatives, such as Fitzgibbon et al.’s least square fitting algorithm, are prone to numerical instability when the ellipse is nearly circular or if sensor data contains correlated noise, which cannot be excluded in our setup [39]. Other options, such as geometrical fits typically rely on iterative refinement rendering such approaches resource-intense and thus unfit for mobile contexts. In a performance comparison, Hemsi identified Halír and Flusser’s algorithm as the most efficient (with a complexity of $O\left(N\right)$) in a performance review of ten different ellipse-fitting algorithms [40], which further motivates our choice.

3.3.4. Ellipse Major Direction

We determine the direction in which the rotating magnet is located along the major axis by exploiting the asymmetry of the magnetic field. For each rotation, the captured signal is an ellipse, even for slightly skewed fields. The ellipse’s extremities correspond to the north and south poles of the field, respectively (see Figure 6). With constant rotational speed, this means that the semi-rotation from north to south pole in a clockwise rotation takes longer to complete than its opposite side. With this information, the north and south poles can be distinguished. We tested two algorithms:

1.

Distance-based: for each complete ellipse, we calculate the sequence of data points with the maximum average distance.

2.

Time-based: each sensor value is timestamped and we calculate the time difference between two semi-rotations.

To exploit the distance between sensor values for determining in which direction along the major the prototype is located, we calculate a moving average of Euclidean distances between sensor values in the ellipse data (Equation (2)).

$$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$$

(2)

The highest average distance is then used as a vector in the cross-product with the ellipse’s major axis to determine the direction of the magnet. For the time-based algorithm, the closest sensor values (x1–x4, Figure 9) to the ellipse major are determined through a complete iteration of the ellipse data points, evaluating the cross product between each centered datapoint to the major axis (Figure 9). We calculate the signed distance for each datapoint and its predecessor.

If the signed distances are not either both positive or negative, one is positive and the other is negative, indicating a crossing between them. The timestamps between these timestamps are then interpolated and used as virtual data points to measure time differences.

The current software privileges a distance based direction detection (due to its better performance). The complete algorithm is illustrated in Figure 10: For each new sensor reading of the mobile device, the raw x–y magnetometer values are added to a buffer and smoothed through a low-pass filter (moving average) to reduce high-frequency noise. An ellipse is then fitted to the filtered buffer, and its Sampson error is computed to estimate the reliability of the fit. A second filtering step keeps only stable ellipse estimates, which are used for binary direction detection. Detected directions are accumulated in a direction buffer, where entropy peaks, corresponding to noise, are identified and filtered. The resulting direction vectors form the basis for estimating the ellipse’s major-axis length, which is transformed into a distance estimate, based on the initial user calibration that maps distance to recorded field strength at known positions. The final position is obtained by converting the direction–distance pair from polar to Cartesian coordinates.

4. Evaluation Methods

To evaluate the feasibility of the proposed SPART-MAR prototype, we designed a set of experiments aimed at validating its robustness and applicability, while also analyzing possible constraints that may arise in real-world deployment. Experiments were conducted in a multi-laboratory effort in two distant laboratories, one in Zagreb, Croatia (Lab 1), and the other in Copenhagen, Denmark (Lab 2).

4.1. Prototype and Experimental Setup

Two SPART-MAR prototypes were assembled and tested independently in both laboratories, Lab 1 and Lab 2. Both prototypes share the characteristics described in Section 3, each using four bar-shaped neodymium magnets arranged in an asymmetric arc, mounted on a DC motor operating at 60 RPM. The rotating structure for both prototypes was encased in a lightweight LEGO-based housing (see Figure 5). Experiments were conducted independently using a grid of 5 × 5 cm squares.

Evaluation data were collected using smartphones with built-in magnetometers. The following devices were used:

• Samsung S21 Ultra running Android 15, dimensions 165.1 × 75.6 × 8.9 mm.
• Samsung A53 running Android 14 smartphone, dimensions 159.6 × 8.1 × 74.8 mm.

Prior to the experiments, the calibration phase (as described in Section 3.3.1) was performed on smartphones to establish the relationship between magnetic field strength and the device’s distance from the rotating magnet.

The calibration process is part of the developed Android application, with instructions guiding the user to place the device at known distances to the magnet, and enter this information (see Figure 11 center). For calibration, smartphones were positioned on a printed grid, with points spaced at 5 cm intervals relative to the prototype. Based on at least three known reference positions (i.e., 10, 15 and 20 cm), the software performs a logarithmic regression to determine the calibration parameters. The same application was later used for measurements and data collection on smartphones. (see Figure 11 right).

4.2. Experimental Procedures and Data Collection

In the experiments carried out in the Zagreb laboratory (Prototype 1 in Lab 1), two configurations were tested, with a 5 mm KAPA^®fix grid placed both below and above the prototype, positioning the magnets 1 cm below the board, resulting in a distance of approximately 1.5 cm from the mobile device. In the experiment conducted in the Copenhagen laboratory (Prototype 2 in Lab 2), the calibration and measurement grid, printed on paper, was positioned beneath the prototype. In each setup, smartphones were systematically moved across the grid in 5 cm increments along both the x- and y-axes, mobile devices aligned to the grid (see Figure 12).

The following tests were conducted:

• Positioning accuracy testing: comparing the ground-truth distances recorded on the measurement grid with the corresponding values estimated by the application (evaluating criteria 1—Accuracy).
• Distance testing: assessing the maximum operational range of the prototype relative to the device (criteria 3—Minimum operating range).
• Obstruction testing: placing the prototype beneath a 5 mm KAPA^®fix board.

Data was collected from the Android application, displaying calculated position, major length and angle. Measurements were conducted until the data displayed in the application and the fitted ellipse became unstable, at which point further readings could no longer be reliably obtained due to instability in the numerical output. All measurements were obtained with smartphones in portrait configuration in the quadrant south-east of the prototype (see Figure 12).

5. Data Analysis

We measured the accuracy and stability of the algorithms in both laboratories. The following sections report the data from these experiments.

5.1. Accuracy

We measured accuracy by positioning smartphones at 5 cm intervals on a grid, reporting actual and measured position, where readings were stable. Figure 13 shows the measured positions and calculated Euclidean distances to the actual position, adjusted for sensor offsets (prototype 1: 0.5 and 0.3 cm, prototype 2: 1.1 and 2.3 cm) in the smartphones.

We took into account only positions that were measured by both devices, since prototype 1 shows a larger coverage (e.g., with readings for position 30, 30) than prototype 2, where readings became unstable for closer distances).

One error observation for prototype 2 (14.21 cm) was identified as an extreme outlier based on its excessive influence on the standard deviation and was thus excluded from the final descriptive statistics and modeling, as it likely represents a mistake during manual data collection. We thus report accuracy for $n=43$ observations. Prototype 1 showed a higher mean error of $e_1=1.77$ cm, a standard deviation $SD=1.21$, a 95% confidence interval for the true mean error of $CI=[1.40,2.14]$ and an interquartile range $IQR=1.81$ (calibration curve fitting error of 1). Prototype 2 has a lower average error of $e_2=1.50$ cm, $SD=0.89$, a 95% confidence interval of $CI=[1.23,1.77]$ and an $IQR=1.55$ (calibration curve fitting error of 0.95). Figure 14 shows the error distributions as histograms.

We further conducted a paired t-test to verify whether the mean differences are statistically significant. There is no statistically significant difference between both, with $p\approx 0.30$. We further investigated whether the error correlated to distance. As Figure 15 (left) shows, no correlation exists between error and distance. However, a clear correlation between error and angle can be observed for both prototypes (Figure 15 right).

We conducted a Pearson correlation test for both prototypes. In both cases, a statistically significant negative correlation between error and angle was confirmed ($r_1=-0.81, r_2=-0.78$ and $p_1, p_2=<0.001$). Errors are highest close to 90 degrees and decrease for higher angles.

5.2. Stability and Range

In both setups, we noticed issues with the direction algorithm at increasing distances. All experiments were conducted with the distance-based algorithm. Femm simulations suggest that the magnet shape’s impact on its field decreases over distance (for longer distances, the shape resembles a symmetric magnet). During our experiments, this phenomenon became apparent with increasingly unstable readings over distance. The current algorithm makes use of a buffer of 100 direction vectors to stabilize angle measurements. Thus, a flipping direction vector decreases the average vector in length. Consequently, we reported these occurrences in Figure 16 through three labels: stable, slightly unstable and unstable. During “slightly unstable” readings, the effect was observable occasionally, in unstable readings, the effect heavily impacted the readability and stability of readings.

From our measurements, we can deduct an operational range of 25–30 cm for the current prototype. In terms of refresh rate, the buildup of stable measurements occurs after 1–2 s after a position change due to buffer updates.

5.3. Additional Experiments

Beyond experiments to determine stability and accuracy conducted in both laboratories, the following experiments were conducted separately.

5.3.1. Non-Obstructive Performance

The potential of the prototype for its use beneath the surface (instead of on top of the surface) was tested in Lab 1. The prototype was, in this configuration, positioned 1 cm under a 5 mm KAPA^®fix surface (see Figure 17).

While the surface directly above the prototype did not yield stable readings, the mean Euclidean error for the remaining positions increased, compared to on-surface readings by 0.4 mm to 2.16 cm for the “beneath-surface”-configuration, with $SD=1.348,IQR=1.69$ and $CI=[1.79,2.52]$. Figure 18 shows the results for each position, depicting the on-surface measurements in blue and the beneath-surface measurements in orange.

5.3.2. Angle and Length Errors

Since the errors in measurements, depicted in Figure 13, Figure 14, Figure 15 and Figure 16 and Figure 18, showed a pattern, confirmed by the error-to-angle correlation in Figure 15, Lab 2 investigated these errors as a product of length and angle calculation in the algorithm. Having identified the position of the magnetometer within the smartphone, we measured the angle error for different distances along a straight line, with the smartphone oriented at the same angle relative to the prototype. We did so for three configurations: landscape, portrait and a 45° angle.

While the error decreases from the negative spectrum towards 0 over distance for the phone facing the prototype with its long side (i.e., landscape), the portrait and 45-degree configurations both initially decrease in error before increasing beyond a distance of 40 cm (see Figure 19). The different curves illustrate the differences in penetration of the magnetic field depending on its relative angle to the mobile device, adding to the complexity of an additional calibration procedure to establish a corrector function for this distortion.

We further conducted angle and distance measurements on an increased set of measurements (200 measurements per position), allowing for an estimation of the variability of measurements. Figure 19 shows this variability over distance and per configuration. While configurations where the phone faces the prototype, either horizontally (i.e., portrait) or laterally (i.e., landscape), closely align, this is not the case for the phone in a 45-degree configuration (see Figure 19). Instead, the 45 degree curve is slightly delayed to portrait or landscape configurations and more unstable compared to the former, adding to the difficulty to implement a corrector function that can both compensate angle and length error through a simple calibration procedure. Standard deviation for these measurements also is also higher compared to landscape and portrait mode (see Figure 19). Concerning angle error stability, for all configurations, the angle variability increases to plus or minus 2 degrees (Figure 19: bottom), while major length stays stable until 50 cm distance (with the exception of the 45 degree configuration). A potential corrector function to increase accuracy could thus rely on distance measurements, accounting for the different errors in angle over distance.

5.3.3. Investigations into Direction Stability

Following the observation of stability issues with increasing distance (Section 5.2), we investigated the issue and attempted a remediation strategy with the time-based algorithm described in Section 3.3.4. With a complementary binary buffer to capture a history of ellipse major directions, we calculated and traced buffer entropy for space and time-based algorithms at a distance of 50 cm (see Figure 19).

We expected increased stability from the temporal algorithm, but as Figure 20 illustrates, flips occur three times as often in the same time interval (smartphone at rest). We can thus conclude that a temporal algorithm does not improve performance or solves the issue of occasional flips in the major direction length calculation. The results also show that the overall data is stable (for the spatial algorithm), and that reported instability of readings is due to isolated peaks in the data, even for distances that were not reportable in Section 5.

5.3.4. Alternative Magnetic Position Indicators

Given the limitation of distance impacting detection of asymmetry of the magnetic field, we investigated an alternative approach. Instead of rotating the magnet in two dimensions, we inclined a symmetric magnet so that the north pole points slightly upwards and the south pole downwards (5 degrees). We then measured the magnetic field strength along an additional z-axis over distance to investigate whether the Z-data could provide a more reliable indicator, while simultaneously removing the necessity for angle correction due to field asymmetry. We used the 10 × 40 × 5 mm bar magnets as a combined, symmetric magnet in this setup, with an inclination of 3 degrees of the magnet towards the surface.

As Figure 21 shows, the Z-measurements can be used to distinguish magnetic poles reliably for distances up to 30 cm, before signals become noisy. For close distances such as 20 cm, the range between peak and valley is typically 1 μT, which decreases to four or fewer μT for distances around 30 cm, with 5 μT being the typical noise range of smartphone magnetometers. Given this limitation, we do not recommend this approach for future developments.

6. Discussion

The different conducted experiments assessed the current prototype’s potential for the SPART criteria mentioned in Section 2. The assessment overview is presented in

Table 1. Assessment of SPART-MAR prototype according to SPART criteria.

Accuracy	The current error is situated between 1.3–1.6 cm across two different prototypes and smartphones, and thus does not fulfill the threshold of 1 cm.
Fluidity	The current update rate, only at rest, is below 1 Hz and thus not sufficient.
Minimum operating range	The current prototype operates reliably in a radius of 25 cm, of which the inner ring of 5 cm is not accessible due to sensor saturation and obstruction, leaving an operational area of 1885 cm2. The required surface of an A3 is 1247 cm2. Although the covered surface is larger, it is not rectangular. The requirement is thus partly met.
Mobility	The prototype is mobile (weight is around 500 g), and the requirement is thus met.
Affordability	Components can be ordered for less than €10.
DIY & Reparability	The prototype can be built nearly entirely with LEGO components. Soldering can thus be avoided entirely.
Multi-device support	Multiple devices can be located simultaneously.

.

While the requirements of accuracy, fluidity, and operating range are currently not, or only partly met, this first implementation of SPART-MAR is proof of the applicability of the concepts we employed for implementing a wireless, DIY tracking system for off-the-shelf mobile devices. The stability of angle and length (Figure 19) shows potential for extending the operating range to 50 cm with a potential of improved accuracy with an angle-corrector function. Fluidity can further be improved through the use of partial ellipse detection, allowing for a potential increase in fluidity and use in motion. Performance under the surface led to decreased accuracy (−0.6 cm), but localization remains possible, opening a perspective for integrated solutions without the need of the spinning device on the same level as the mobile devices.

To clearly outline the contribution of SPART-MAR within the family of SPART prototypes, and to contextualize the findings reported in this study, we provide a comparison with previous implementations: SPART-ME and SPART-AC. These earlier prototypes differ substantially in their underlying localization techniques, hardware requirements, and practical usability.

Table 2. Comparison of SPART-MAR with prior SPART prototypes.

Criterion/System	SPART-MAR (This Work)	SPART-ME (Mechanical)	SPART-AC (Acoustic)
Accuracy	1–3 % (on axis)	≤0.5 cm	Highly variable
Fluidity / Update rate	∼1 Hz (device at rest)	Smooth, real-time	None, updates only at rest)
Operating range	25–30 cm radius	Full A3 surface	Full A3 surface
Mobility	High (compact; <500 g)	Moderate (rigid support required)	None (20 kg)
DIY / Reproducibility	Very high (BLE/OVD print; <10, minimal electronics)	Moderate (complex assembly and calibration)	Low (microphone array and circuit cost €100)
Multi-device support	Yes	Limited (one thread per device)	No (single device)
Dependence on external hardware	None (uses magnetometer only)	External Bluetooth chipset	Microphone array, sound processing on external, Bluetooth enabled hardware
Overall contribution	First wireless SPART implementation; low-cost and portable	High accuracy and fluidity	Conceptual proof-of-concept, not yet practical

summarizes key performance criteria, highlighting where SPART-MAR advances the state of the art and where its current limitations remain.

As shown in Table 2, SPART-MAR is the first SPART prototype to achieve localization without physical coupling to the device, while also meeting the goals of mobility, low-cost, and DIY reproducibility. At the same time, limitations in fluidity and long-range stability distinguish it from the mechanical prototype (SPART-ME), underscoring the need for continued refinement of angle detection and signal processing in future iterations.

Positioning SPART-MAR within the broader landscape of interaction techniques reviewed in Section 1 highlights both its novelty and its current limitations. In contrast to camera-based AR approaches (e.g., handheld AR systems and multi-device AR setups), SPART-MAR avoids issues of device appropriation, fatigue, and reduced socio-cognitive availability by enabling on-surface dynamic peephole interactions that keep the device stable and accessible to all group members. Compared with spatially aware systems like Huddlelamp [29], whose depth-camera infrastructure offers high accuracy but poor mobility and high cost, SPART-MAR provides a portable, low-cost alternative that does not require a data connection with external tracking hardware. Moreover, unlike previous peephole implementations that rely on inertial sensing or 3D tracking, SPART-MAR leverages a simple rotating magnet to support real-time spatial mapping on 2D surfaces, aligning with the original SPART vision of augmenting printed media with mobile devices. At the same time, the findings show that SPART-MAR does not yet match the accuracy and fluidity of the mechanical SPART-ME prototype or of high-end camera-based systems, indicating that further work is needed to refine its stability at larger distances and improve its responsiveness during device movement.

It should be highlighted that the current prototype is a proof of concept. Given its limitations in terms of fluidity, it requires further improvements before authentic usability study can be conducted, and comparisons to prototypes like SPART-ME (that have proven successful collaboration supports) can be drawn. Issues related to heterogeneous penetration rates depending on angle further complicate a future calibration process: while the angle corrector function for distance can be integrated without complementary steps in the current procedure, generating a profile of the device that reflects its different penetration properties by angle and locating the magnetometer within the device are likely to lead to a more complex calibration setup.

Relatedly, generating a strong magnetic signal is likely to perturb existing soft-iron configurations within smartphones, used for other applications (navigation, etc.), likely requiring a reconfiguration after the use of SPART-MAR.

Finally, for ellipse signals at around 50 cm, we noticed substantial angle distortions (3 degrees) correlated to the smartphone’s activity, like processor tasks or wireless communication (e.g., Bluetooth).

7. Conclusions and Future Work

In this paper, we presented a proof of concept for a localized 2D tracking system called SPART-MAR. The developed prototype has been tested in a series of multi-laboratory experiments, evidencing its potential to meet requirements for a substitute of large and expensive tabletop devices (accuracy, fluidity, operating range, mobility, affordability, reparability, and multi-device support), and this first version’s current limitations.

These limitations provide directions for continued development. Future work will primarily focus on algorithmic optimization, including partial ellipse detection to increase update rates, angle-correction modeling to increase range beyond 25–30 cm, and sensor-fusion strategies to suppress entropy spikes during direction estimation, extending effective range to 50 cm (covering a potential, circular surface of 7600 cm² or 6 A3 sheets). We also plan to extend the system toward 4DOF (four degrees of freedom) tracking by leveraging the ellipse center as an indicator of device rotation, which would improve usability by removing the need for fixed device alignment.

Future work will also include systematic performance evaluations across a wider range of environments, such as different surface materials, classroom conditions, and mobile devices, to more robustly assess the generalization of SPART-MAR.

Finally, integration with mobile AR frameworks and dissemination to educational settings will guide the next stage of investigating SPART-MAR’s potential to support collaborative learning on augmented physical surfaces.