1 Introduction↩︎

The interaction between magnetic sources has unveiled intriguing characteristics, such as the ability to transmit motion without physical contact and provide localization support even when direct view is not available [1]–[3]. Consequently, the field of magnetically-driven medical technologies, particularly those involving the interplay between magnetic sources and the human body, has seen notable advancements, with several significant achievements already documented [1]–[3]. Permanent magnets have been widely used in the development of medical systems for purposes such as localization, actuation, screening, and drug delivery [1]–[3]. In this context, developing an effective magnetic model is crucial for designing medical devices that utilize permanent magnets for actuation or as a driving force. Despite the potential benefits of magnetic technology, its application in rehabilitation robotics has been restricted [1], [4], [5].

Rehabilitation robots have become increasingly important in helping individuals recover from sensorimotor impairments, predominantly caused by strokes, which are a leading cause of death and long-term upper limb motor disabilities [6]. Physical therapy is essential for stroke survivors to regain motor control, but traditional methods often fall short due to their intensive demands [7]. As a result, robotic systems for upper limb rehabilitation have been developed to make the recovery process more efficient, allowing patients to independently practice and strengthen their abilities [8]–[12]. These robots, designed for exercises ranging from finger manipulation to shoulder movements, have shown promise in studies comparing them to conventional therapies [7], [13], [14]. However, their wider adoption is hindered by factors like high costs and complex designs, which limit accessibility [8]–[10]. Additionally, the need for simple, user-friendly interfaces is critical for users with severe impairments [9]. To facilitate their use at home, rehabilitation robots must be portable, safe, and easy to operate, overcoming both physical and financial barriers to access [9].

Upper limb rehabilitation robots are primarily designed as either end-effector-based or exoskeleton-based systems [8]–[11]. Exoskeleton-based robots, with multiple degrees of freedom, allow for a wider range of movements due to more connection points [8]. In contrast, end-effector-based robots are predominantly adopted due to their simplicity, ease of production and control, and adaptability to different arm sizes of subjects [9]. Additionally, they have the lowest cost and the greatest potential for commercialization [9], [15]. The study [16] has proven the effectiveness of end-effector-based devices over exoskeleton-based robots. These end-effector robots are further categorized into three types: robots for manipulation [5], [17]–[23], for reaching [24], [25], [25], [26], [26], [27], [27], [28], [28]–[39], [39]–[52], and for both manipulating and reaching [53], [54]. While robots designed for reaching activities are often prioritized in rehabilitation due to their focus on training upper body movements prior to hand manipulation, they also cater to a wide range of patient needs [55]. Starting with a simple design of just one degree of freedom (DOF) for those with severe conditions, these robots can be adapted to include up to three DOFs, facilitating movement training across three axes [9], [55].

A few of the end-effector-based platforms for reaching with different designs are as follows: MOTORE [45]–[47] is a portable mobile robot, equipped with trans-wheels, three DC motors to help patients follow preset circular routes in a graphical interface. Campolo et al. [35], [48], [48] designed a planar robot with simple mechanical properties based on H-shape cabled differential transmission using two DC motors for arm treatments. “Braccio di Ferro” [49], [50] is a robotic arm developed based on parallelogram linkage using two motors.

In developing these platforms, two primary challenges often emerge: the potential safety risks when users directly interact with system components, and the systems’ failure to accommodate the human hand’s weight. Additionally, ensuring smooth actuation and motion remains crucial. We hypothesized that integrating magnetic technology as the actuation method could address these concerns. Therefore, we created a planar rehabilitation robotic system to evaluate the viability of our innovative magnetic actuation mechanism. This mechanism is designed to transmit forces seamlessly from the driving elements (such as motors) to the following elements (like the hand wrist), ensuring a smooth and safe interaction. To the best of our knowledge, the use of magnetic technology for actuation in rehabilitation robotics is unprecedented.

To study the system model and the magnets’ behavior, we opted to use the Kalman Filter. The utilization of the Kalman filter across various studies highlights its pivotal role in enhancing the control and estimation capabilities of robotic rehabilitation systems [56]–[64]. From its integration into discrete control schemes for disturbance estimation in upper limb exoskeletons, such as RehabRoby [64], to its application in sensor fusion for reducing the need for multiple sensors while maintaining control quality [59], the Kalman filter proves essential for accurate state and parameter estimation amidst noisy measurements. Moreover, the Kalman filter’s adaptation for improving joint position estimation in Kinect systems [56] and for precise IMU-based motion tracking in virtual reality rehabilitation setups [60] showcases its critical role in developing more efficient, responsive, and personalized rehabilitation technologies.

Therefore, in our study, we employed the Kalman filter not only to validate the system dynamics but also to monitor the magnets and compensate for any disturbances. Our results reveal potential avenues for future use of magnetic technology in rehabilitation robotics.

2 Platform Development and Characterization↩︎

2.1 Hardware architecture↩︎

Our platform features a simple mechanical design operating on a single axis, as illustrated in Figure 1. It is driven by a Teknic DC motor, with its rotor attached to a drive pulley, moving a belt looped around two fixed pulleys. This setup enables a linear slider, carrying a permanent magnet as the end-effector, to move. To track the end-effector’s position, we’ve installed five photo interrupter sensors along the slider, linked to an Arduino board for processing. Sensors at the slider’s ends also act as safety stops to prevent overruns.

For safety and to counterbalance the hand weight, the mechanism is enclosed in a plexiglass casing, ensuring magnet separation and minimizing direct contact. Friction is reduced via two frictionless rings, ensuring smooth motion. The motor positions the end-effector at the start, with subsequent movements defined by the technician through a GUI control panel.

2.2 Software↩︎

The software for our system, highlighted in Figure 2, is crafted using Python 3.11 and incorporates a graphical user interface (GUI) for visual feedback and operational control by technicians. It features a turtle icon as a cursor to represent the end-effector’s location. This GUI allows for setting and adjusting the end-effector’s path toward a defined target. Communication with the hardware is managed through PySerial for serial command exchanges. The software supports inputting data like user or trial numbers and defining trial trajectories. It also visualizes and presents trial data for the technician’s analysis.

2.3 Dynamic Model of the System↩︎

In this case study, we focused on cylindrical permanent magnets. Typically, the interaction between such magnets is considered through the lens of magnetic dipoles. However, given the specific shapes and distances of our magnets, we employed a more detailed analysis. A free-bodied diagram is shown in Figure 3, representing our novel actuating mechanism and forces acted upon the magnets.

The following formula calculates the attractive magnetostatic force between two laterally displaced cylindrical magnets according to [65], [66]:

\[A = \frac{1}{d^2} + \frac{1}{(d+2h)^2} - \frac{2}{(d+h)^2}\]

Then, the force can be expressed as:

\[F_{\text{mag}, x} = -\frac{\pi K_d R^4}{2} \Bigg[ A \\ - \frac{3}{2}(p_1 - p_2)^2 A^2 \Bigg] \cdot \text{atan}(\alpha)\]

In this formula: \(\mu_0 = 4\pi \times 10^{-7}\) is the magnetic permeability, \(M\) is the magnetization of the magnets, \(d\) is the separation distance, \(h\) is the height (thickness) of the magnets, \(R\) is their radius, \(\alpha\) is the angle between the magnets on a horizontal plane, and \(p_1\) and \(p_2\) are the positions of the bottom and top magnets.

As for the friction analysis, for the top magnet, which exhibits a stick-slip motion, the friction is best described using the Stribeck model according to [67]:

\[F_{\text{fric},1} = \left[F_c + (F_s - F_c) \cdot e^{-\left(\frac{v}{v_s}\right)^2}\right] \cdot \text{sgn}(v) + K_v \cdot v\]

Where: \(F_c\) is the Coulomb friction force, \(F_s\) is the static friction force, \(v_s\) is the Stribeck velocity, and \(K_v\) is the viscous friction coefficient.

Conversely, the bottom magnet, which changes direction but does not display stick-slip motion, is primarily subject to viscous friction. Although there is some static friction between the bottom magnet and the glass surface, it is negligible and offset by the motor force. The system is modeled as a two-degree-of-freedom (2-DOF) system and the kinematics of the two magnets are the output, the motor force is the input. The top magnet’s displacement correlates with the bottom magnet’s displacement, plus an offset.

For the bottom magnet, we consider its acceleration \(a_1\) in relation to the net force acting on it. According to Newton’s law:

\[F_{\text{net}} = m \cdot a_x\]

The net force for the bottom magnet (\(F_{\text{net},1}\)) and for the top magnet (\(F_{\text{net},2}\)) is calculated as: \[F_{\text{net},1} = F_{\text{motor}} - F_{\text{mag}, x} - F_{\text{fric},1}\]

\[F_{\text{net},2} = F_{\text{mag}, x} - F_{\text{fric},2}\]

In these formulas, \(F_{\text{motor}}\) is the force applied by the motor. The selection and adjustment of hyper-parameters were carried out manually, guided by benchmarks from similar studies to set appropriate values.

2.4 Static and Dynamic Characterization↩︎

In our investigation, we focused on the synchronized movement of cylindrical permanent magnets within a rehabilitation platform, as visually represented in Figure 4. Our primary aim was to confirm that the magnets not only moved in unison but also remained attached to each other throughout the operation of the platform, even under the various forces exerted by patients, such as hand movements and the inherent weight of the limb.

To ensure the reliability and safety of the system, we embarked on a comprehensive characterization of both the static and dynamic behaviors of the magnets. In the dynamic mode of our experiment (Figure 4-a), we incorporated sophisticated measurement tools to precisely gauge the movement synchronization between the magnets. The bottom magnet’s motion was tracked using a GHB38 rotary encoder attached to the platform’s right shaft, providing detailed insights into its speed and positional changes. Simultaneously, the movement and distance of the top magnet were monitored using a HiLetgo VL53L0X time-of-flight distance sensor. This way, we could simulate patient interaction with the system in passive rehabilitation. This dynamic testing was critical to ensure that any lag or difference in the magnets’ movements did not negatively impact the therapeutic effectiveness or the user experience of the rehabilitation process.

Shifting to the static analysis (Figure 4-b), the bottom magnet was fixed in place, creating a controlled environment to observe the movement of the top magnet. This setup was subjected to different levels of force applied through calibration weights, mimicking the range of forces a patient might exert during rehabilitation exercises. The goal was to establish the threshold force that could potentially cause the magnets to detach. This threshold was crucial as it needed to be high enough to prevent unintentional detachment during normal use, yet low enough to not pose excessive resistance to the patient.

An important aspect of our study was considering only the transitional forces (represented by gravitational force here) and normal force in these tests, aligning with the common scenarios encountered in passive rehabilitation. However, we also acknowledged the possibility of additional force resistance that patients might experience due to limited limb functionality. Such resistance could potentially introduce an increased offset in the magnet synchronization, presenting a scenario that our system needed to accommodate to maintain its efficacy and safety in a real-world rehabilitation context.

3 Algorithm Development↩︎

To investigate the system, monitor the behavior of the magnets, and effectively address the challenge of keeping the magnets in our rehabilitation device from detaching due to unsolicited forces, we engineered a resilient closed-loop algorithm as depicted in Figure 5. The algorithm’s role is to monitor and manage the distance between the magnets as they travel along the platform. It’s programmed to detect any increasing gap that could lead to detachment and to intervene accordingly. This proactive adjustment is crucial for smooth and uninterrupted patient interaction with the system.

For this intricate task, we chose to implement the Extended Kalman Filter (EKF) technique, which is particularly effective for such systems displaying nonlinear dynamics [68]. The EKF, an enhanced version of the Kalman Filter (KF), excels in linearizing nonlinear systems at each iteration, enabling the application of standard Kalman filter equations. This method not only ensures the magnets’ gap is efficiently managed to prevent detachment but also provides us with deeper insights into the physical principles governing the system, enriching our understanding of its mechanics and enhancing its effectiveness in rehabilitation exercises.

3.1 state-space representation↩︎

In order to simplify the process of system dynamics (Figure 5, left side) we use state-space representation which is a mathematical model of a physical system as a set of input, output, and state variables related by differential equations [69]. In state-space representation, a system is described by two equations:

1. State Equation: Describes how the current state of the system (a vector of state variables: x) evolves based on the current state and the input to the system (motor force in our case). It is usually expressed as first-order differential equations. As shown in Figure 5, it is the fundament of our algorithm.

2. Output Equation: Relates the current state of the system to the output.

Based on our system and the formulation described above, we have 4 state variables which are the positions and the velocities of the magnets. Our output is either the position of the two magnets or the position of the bottom magnet only. Hence, our state equations are as follows:

\[\Dot{x}_{\text{1}} = v_{\texttt{1}}\]

\[\Dot{v}_{\text{1}} = \frac{F_{\text{motor}}}{m_{\text{1}}} - \frac{F_{\text{mag,x}}}{m_{\text{1}}} - \frac{F_\text{fric,1}}{m_\text{1}}\]

\[\Dot{x}_{\text{2}} = v_{\texttt{2}}\]

\[\Dot{v}_{\text{2}} = \frac{F_{\text{mag,x}}}{m_{\text{2}}} - \frac{F_\text{fric,2}}{m_\text{2}}\]

\[x= \begin{pmatrix} \Dot{x}_{\text{1}} \\ \Dot{v}_{\text{1}} \\ \Dot{x}_{\text{2}} \\ \Dot{v}_{\text{2}} \end{pmatrix}\]

3.2 Extended Kalman Filter↩︎

The Extended Kalman Filter (EKF) is an advanced algorithm used for estimating the state of a dynamic system, particularly when the system’s behavior is nonlinear [68]. It extends the concepts of the standard Kalman Filter (KF) to handle nonlinearities by linearizing them at each step. The EKF operates in two main phases: the prediction phase and the update phase. These phases work in a cycle, continuously updating the system’s state estimate as new data comes in. During the prediction phase, the EKF forecasts the future state of the system based on the current state and any control inputs. This step involves two key formulas: The state prediction formula [69]: \[\hat{x}_{k|k-1} = f(\hat{x}_{k-1|k-1}, u_{k-1})\] Here, \(\hat{x}_{k|k-1}\) is the predicted state at the next time step (k), based on the current state (\(\hat{x}_{k-1|k-1}\)) and control input (\(u_{k-1}\)). The function \(f(\cdot)\) represents the nonlinear state transition of the system. The covariance prediction formula [69]: \[P_{k|k-1} = F_{k-1} P_{k-1|k-1} F_{k-1}^T + Q_{k-1}\] This equation predicts the covariance of the state estimate, where \(P_{k|k-1}\) is the predicted covariance, \(F_{k-1}\) is the Jacobian matrix of \(f(\cdot)\) with respect to the state, and \(Q_{k-1}\) is the process noise covariance.

In the update phase, the EKF adjusts the predicted state using the new measurement data: The Kalman gain is computed as [69]: \[K_k = P_{k|k-1} H_k^T (H_k P_{k|k-1} H_k^T + R_k)^{-1}\] This equation calculates the Kalman gain (\(K_k\)), which determines how much the measurements should influence the state estimate. \(H_k\) is the Jacobian of the measurement function, and \(R_k\) is the measurement noise covariance. The state update formula is [69]: \[\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (z_k - h(\hat{x}_{k|k-1}))\] Here, \(\hat{x}_{k|k}\) is the updated state estimate, \(z_k\) is the actual measurement, and \(h(\cdot)\) is the nonlinear measurement function. The covariance update formula is [69]: \[P_{k|k} = (I - K_k H_k) P_{k|k-1}\] This updates the estimate covariance (\(P_{k|k}\)) for the next iteration.

The EKF repeats these predictions and updates steps iteratively. Each cycle refines the state estimate by considering the latest data and the system’s dynamics, allowing the EKF to track the state of a nonlinear system accurately over time. Overall its flow is shown in Figure 5.

3.2.1 System Observability↩︎

Observability plays a crucial role in the effectiveness of the Kalman filter [70]. For the Kalman filter to perform accurately, it requires that the system it’s applied to be fully observable, meaning every aspect of the system’s internal state can be determined based on its outputs. Without observability, the Kalman filter’s ability to accurately predict and adjust the system’s state is significantly diminished, emphasizing its importance in dynamic systems monitoring and control. Adding to this, the concept of partial observability addresses situations where it’s impossible to fully discern the system’s state from its outputs. In such cases, despite the challenges, the Kalman filter can still be adapted to provide the best possible estimate of the system’s state, showcasing its versatility in dealing with the intricacies of real-world applications where complete information may not always be accessible.

3.2.2 Offline tunning of EKF↩︎

Offline tuning of the Extended Kalman Filter (EKF) involves adjusting and optimizing the filter’s parameters based on historical data rather than in real-time. In offline tuning, the EKF’s performance can be meticulously analyzed and refined, as various parameters such as process noise covariance (\(Q\)), measurement noise covariance (\(R\)), covariance (\(P\)), and initial state estimates can be fine-tuned to minimize the estimation error. By evaluating the filter’s accuracy against known outcomes, one can iteratively adjust the EKF settings, leading to a more reliable state estimation when the system is deployed in a live environment. This method of tuning is especially beneficial for systems that operate in predictable environments or have repeatable operations.

In our system, we manually tuned the \(Q\) (process noise covariance) and \(R\) (measurement noise covariance) matrices through a trial-and-error process. This iterative method was suitable due to the system’s operation along simple, predefined trajectories and distances, which are specific to the exercises it performs. We began by initializing the \(P\) matrix (error covariance matrix) and then, through the iterations of the algorithm learning, its values were refined. These optimized values of the \(P\) matrix were subsequently employed in the closed-loop Extended Kalman Filter (EKF) algorithm to enhance its performance. Additionally, we set the initial states of the system based on the predetermined starting point locations, ensuring that the algorithm began with a contextually accurate representation of the system’s status.

3.2.3 Closed-loop state estimation using EKF↩︎

In our setup, the Extended Kalman Filter (EKF) was employed to ascertain the positions and velocities of the two magnets. This was done by applying the EKF algorithm to the system’s established formulation. First, we took into account the positions of the magnets, measured by the laser distance sensor (top magnet) and the encoder (bottom magnet) as our measurements, the motor force as the control input variable, and the positions and velocities as our state variables. This way the system was fully observable and the EKF played a key role in adjusting for inaccuracies in the data obtained from the laser sensor and validating the system’s dynamic model derived earlier.

As a result of the EKF’s effectiveness, it opened up the possibility of phasing out the need for the laser sensor in the system later on. Hence, further, we considered the position of the bottom magnet as our only measurement, leading our model to be partially unobservable with four state variables as before.

3.2.4 Assesment of EKF estimations↩︎

Root Mean Square Error (RMSE) is a critical metric for assessing the accuracy of EKF estimations, quantifying the average magnitude of errors between the EKF’s estimated states and the actual system states [71]. RMSE is calculated using the formula [72]:

\[RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2}\]

Where \(\hat{y}_i\) represents the estimated values, \(y_i\) the actual values, and \(n\) the number of observations. A lower RMSE indicates higher accuracy, reflecting the EKF’s effectiveness in tracking the dynamics of a system.

3.3 Offset recovery↩︎

To address the potential risk of the magnets becoming detached due to sudden resistance by a patient, we implemented a control strategy using a proportional controller. This controller is designed to manage the speed and direction of the lower magnet, particularly when the gap between the magnets exceeds a pre-set threshold identified in our preliminary studies. During typical therapy sessions, our system operates at a low speed. However, if the upper magnet starts lagging, the lower magnet reverses its direction and increases speed to re-align them. Conversely, if the upper magnet moves ahead too quickly, the lower magnet accelerates in the same direction to maintain the connection. It’s important to note that a minimal offset is necessary for movement. Nevertheless, in situations where the offset becomes excessively large and disruptive, this approach effectively prevents disengagement. Figure 5 shows the flow of our system algorithm.

4 Experimental Protocol↩︎

As a case study and to test the feasibility of the system, we carried out a protocol for data collection: The initial stage, known as calibration, was designed to evaluate and confirm the platform’s functionality in conjunction with human interaction prior to the commencement of the actual data-gathering phase. During this stage, we assessed the participant’s physical capabilities and the range of motion of their hand. This involved taking several physical measurements from the participant to fine-tune the chair’s height, the desk’s height, and the space between the participant’s torso and our robotic platform. Subsequently, we assisted the participant in sitting comfortably in the chair, ensuring an upright posture with their torso aligned against the desk’s edge. Adjustments were made to ensure the heights and distances were optimal for the participants, allowing them to use the robotic platform with ease. Following these adjustments, the participant placed their hand on the armrest, and the device moved their hand along a predetermined path at speeds of 15 and 25 RPMs (2.2 and 3.7 cm/s respectively). This part of the experiment, including the break to switch between speeds, is completed in about 10 minutes. The entire calibration and exercise process is designed to last less than 30 minutes.

4.1 Participants↩︎

In our study, we enlisted five healthy volunteers from the University of Rhode Island, including two males and three females aged 22 to 36 years, all right-handed. None had a history of neurological disorders. Before starting, each participant received a detailed explanation of the experiment and gave written consent. The research methods were approved by the Institutional Review Board (IRB) to ensure they met ethical research standards.

5 Results↩︎

Starting from the static and dynamic characterization of the system, Figure 6 illustrates the offset behavior between two magnets in a drive-follow configuration, to determine the influence of different weights on the detachment threshold at varying speeds. In the left graph, at a low speed of 10 RPM, the increasing amplitude with heavier weights indicates a stronger response leading up to potential detachment. The middle graph, at 20 RPM, shows a more uniform behavior, suggesting a less pronounced impact of weight on detachment at this medium speed. Finally, the right graph, at a high speed of 30 RPM, dramatically showcases that the heaviest weight has a significant peak, indicating a critical point where detachment likely occurs due to the combined effects of high speed and increased weight. According to this article [73], a 2-kg weight pulling force is more than enough. The static experiment showcased the magnetic force’s capability to hold without any movement (Figure 6), maintaining attachment up to a weight of 1000 grams. However, beginning at 1200 grams and beyond, we noticed a gradual increase in separation, leading to complete detachment at a weight of 1700 grams.

Figure 7 illustrates the path taken by the participants over an approximately 4-minute span with speeds of 15 (2.2 cm/s) and 25 (3.7 cm/s) RPMs. The maximum distance achievable was 30 centimeters, and they traversed around 6 meters using this path. Figure 7 presents the distances recorded by the encoder (associated with the bottom magnet/driver: P1) and the laser distance sensor (linked to the top magnet/follower: P2). Figure 7-a and Figure 7-d also reveal the EKF predictions for the positions of the magnets within our observable system, where measurements for both magnets’ positions were available for the model. Lastly, Figure 7-b and Figure 7-e show the performance of our model in a partially unobservable system scenario, where only the position of the bottom magnet was known, leaving us without information on the top magnet’s position. In the figures 7-c and 7-f we can observe the difference between real and estimated values of the bottom and top magnet respectively.

Additionally, the model excels in predicting the position of the top magnet as demonstrated in Figure 7-d, albeit with some spikes observed when the bottom magnet encounters the photo interrupter sensors. These spikes are attributed to mechanical disturbances observed at the start of the movement, which likely contribute to the suboptimal performance of the EKF at these moments, especially within the partially unobservable system context. The RMSE measurements for the EKF estimations of the bottom and top magnet positions were recorded at 0.2 cm and 0.6 cm, respectively. For the partially unobservable EKF, these values were observed to be 0.27 cm for the bottom and 2.06 cm for the top, respectively. These RMSE figures reinforce the conclusions we drew from the visual data. Despite these challenges, the overall performance of both systems in estimating the position of the bottom magnet is nearly exemplary. Finally, Figure 8 shows the performance of our proportional controller for recovering the possible detachment between magnets in case of perturbation.

6 Discussion↩︎

In this research, we developed an innovative, seamless actuation mechanism utilizing magnetic technology to enhance safety and smoothness in motion, particularly for upper limb rehabilitation. This approach, which employs magnets in the actuation system, is novel according to the best of our knowledge, as similar studies have not explored the use of magnets for this purpose [1], [4], [5]. Our observations indicated that the magnets were able to move in unison smoothly during the platform’s operation. Nevertheless, it was crucial to verify that the magnets could move together synchronously without disengaging, taking into account the force exerted by the patient and the hand’s total weight. To achieve this, we conducted a detailed analysis of the system’s static and dynamic behaviors. Our results confirmed that this technique could indeed provide synchronous, smooth, and secure motion, with adequate strength, durability, and compliance for human interaction. Moreover, considering the potential disruptions caused by spasticity or involuntary muscle contractions in post-stroke patients, which could lead to the magnets becoming disengaged and interrupting the exercises, we decided to implement an online closed-loop Extended Kalman Filter (EKF). This approach was not only used to analyze and confirm the accuracy of our system’s model but also to continuously monitor the alignment of the magnets throughout the experiment. By utilizing the EKF, we were able to accurately estimate the positions and velocities of the magnets. This capability was particularly crucial because we lacked direct visibility of the top magnet’s location. However, our algorithm enabled us to precisely track and understand the magnets’ behavior by relying solely on the position of the bottom magnet as the single measurable input for the system.

During our experiments involving human participants, we concluded that our platform’s design naturally compensates for the weight of the human hand, wrist, and arm according to the perspective of the user. This design reduces the risk of joint damage by offering a safer support mechanism compared to other platforms [27], [31], [34], [35], [49] where participants must hold a handle and maneuver the end-effector themselves, bearing the weight of their hand. Additionally, it’s crucial for reaching platforms to concentrate on shoulder rehabilitation, ensuring the wrist remains uninvolved with the movement of the end-effector. Unlike some systems that employ bearings to keep the wrist from bending, our platform utilizes magnets that rotate slightly when moving horizontally. This rotation effectively restricts wrist movement, maintaining a stable wrist angle. However, the degree of this rotation was minimal, only a few degrees, and thus, we considered it negligible in our analysis and formulation earlier. The five human participants expressed satisfaction with the device’s comfort, motion fluidity, and control features. However, it is important to note that the aim of this paper is not to provide a quantitative analysis of our qualitative findings; further research is anticipated to address this aspect.

Lastly, to showcase the ability to manage potential disturbances within our system, we implemented a basic proportional controller. While this controller serves as a proof of concept, it is recognized as the most rudimentary and least effective option. Advanced control strategies could be developed to provide a more refined response to disturbances. This experiment focused primarily on demonstrating our concept of magnetic actuation through what is termed passive rehabilitation, where the robot operates without active patient involvement. However, the real potential of our approach is likely to be realized in assistive rehabilitation scenarios, where both the patient and the robot collaborate to accomplish a task [55]. In such cases, the combination of a sophisticated control algorithm and an electromagnet becomes crucial, potentially highlighting the advantages discussed in our study even further.

6.1 Limitations and Future Work↩︎

Despite the encouraging outcomes, our system has potential areas for enhancement. Initially developed with basic assumptions and limited data, our dynamic model performed well for its intended purpose. However, there exist advanced mathematical techniques that can refine the estimation of its hyperparameters, leading to improved model performance. Initially, photo-interrupter sensors were integrated before incorporating the encoder. The introduction of the encoder offers the flexibility to customize movement spans based on an individual patient’s range of motion, which a therapist and the device can determine. Removing these sensors could also mitigate the mechanical disruptions caused by the start-stop motion, facilitating a smoother operation. Additionally, the encoder presents opportunities for varying speed controls. Considering the use of electromagnets instead of permanent magnets may offer finer motion control and enhanced support. Lastly, while the system is currently designed for one-dimensional movements and targets patients with severe motor impairments, modifications could extend its applicability to those with more advanced motor skills.

7 Conclusion↩︎

In our research, we introduced an innovative concept for powering rehabilitation robots through magnetic technology. For experimental validation, we employed two permanent magnets: one attached to the end-effector of our custom-built platform (termed the driver) and encased in a glass enclosure, and the other positioned above the glass (referred to as the follower). We analyzed the behavior of our system using the Extended Kalman Filter. We concluded that this magnetic actuation method potentially improves the user experience by offering smoother operation, increased comfort, effective compensation for hand weight, and enhanced safety.

Appendix A↩︎

This article contains a video that presents the overall functionality of our platform through the execution of a sample trial.

Acknowledgment↩︎

The authors wish to express their gratitude to Yalda Shahriari for her assistance throughout the development of this platform.

Funding↩︎

The research reported here was supported by the URI Foundation grant on Medical Research and National Science Foundation under award ID 2245558.

References↩︎

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1. S. Ghafoori, A. Rabiee, and R. Abiri were with the Department of Electrical, Computer, and Biomedical Engineering, University of Rhode Island, Kingston, RI, 02881 USA e-mail: (reza_abiri@uri.edu).↩︎

2. M. Jouaneh was with the Department of Mechanical Engineering, University of Rhode Island, Kingston, RI, 02881 USA↩︎

3. *These authors contributed equally to this work↩︎