Rehabilitation Robot Sensor

Of A Low-Cost Non-Backdrivable End-Effector

upper limb rehabilitation device. Front. Rehabil. Sci. 5:1469491.

Distribution Or Reproduction Is Permitted

which does not comply with these terms.

Baiyu Wang, Huaguang Du And Chengkun Zhang

Futronics (NA) Corporation, Pasadena, CA, United States This paper presents GARD, an upper limb end-effector rehabilitation device developed for stroke patients. GARD offers assistance force along or towards a

Physical

therapy sessions.

Non-

backdrivable mechanism with novel motor velocity-control-based algorithms, which offers superior control precision and stability. To our knowledge, this innovative technical route has not been previously explored in rehabilitation robotics. In alignment with the new design, GARD features two novel control algorithms: Implicit Euler Velocity Control (IEVC) algorithm and a generalized

Control

algorithm.

Runtime

complexity for any arbitrary trajectory. The system has demonstrated a mean absolute error of 0.023 mm in trajectory-following tasks and 0.14 mm in trajectory-restricted free moving tasks. The proposed upper limb rehabilitation device offers all the functionalities of existing commercial devices with superior performance. Additionally, GARD provides unique functionalities such

Map

interaction. This device holds strong potential for widespread clinical use, potentially improving rehabilitation outcomes for stroke patients.

Keywords

upper limb rehabilitation, end-effector rehabilitation robot, Assist-As-Needed, motion

1 Introduction

Every forty seconds, an individual in the United States experiences a stroke, and every four minutes, a stroke leads to death. Approximately 7.6 million Americans aged 20 years and older have experienced stroke. As age progresses, the prevalence of stroke increases in both males and females. By 2030, the prevalence rate is projected to increase to 3.9% (1), making stroke a major health concern in the US. Research shows that between 30% to 66% of hemiplegic stroke patients experience limited arm motor function six months after a stroke, with only 5% to 20% demonstrating complete functional recovery (2). These low patient recovery rates are linked to diminished quality of life and increased risks to overall well-being.

Robotic-assisted therapy (RAT) has been found to be a valuable adjunct to conventional physical therapy for post-stroke upper extremity rehabilitation, particularly for subacute stroke patients (3). Compared to conventional physical therapy, RAT allows patients to undergo consistent and repetitive rehabilitation exercises, showing comparable and better outcomes (4). Among those frequently researched RAT devices

End-Effector

rehabilitation devices are particularly popular due to their portability and affordability.

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Meanwhile, the end-effector devices can still deliver patient outcomes that are comparable to more complex systems such as exoskeleton robots (4). The present article introduces the design and control of a low-cost end-effector rehabilitation device. The proposed device and control method shows the potential to offer tailored and long-term adaptive training for stroke patients and extend the device’s product life span.

This paper introduces the Gantry Arm Rehabilitation Device

End-Effector-Based

upper limb rehabilitation robot, as shown in Figure 1E. GARD’s design features a non-backdrivable mechanism, which enhances

Rarely

explored in previous research. The device uses two ball-screw- driven linear actuators for movement along the x and y axes, providing superior stability along the z-axis compared to multi- link robots. Additionally, the direct-drive ball-screw mechanism offers better wear resistance and higher accuracy than belt-driven systems. Our device matches the functionalities of existing

Superior

accuracy.

Additionally,

GARD provides unique features, such as area-restricted free moving, which enhances the flexibility of training protocols. Three main challenges are addressed in this paper: First, a non-backdrivable mechanism presents challenges in constructing an accurate dynamics model. For instance, friction within the transmission can be significantly influenced by various factors such as load, temperature, configuration, lubrication levels, and velocity. To solve this, we develop our control methods based on the motor velocity control and a virtual

Dynamics

model.

For

elusive forces using the PID velocity control and calculates the desired velocity using a virtual dynamics model. Second, the velocity-control-based trajectory-following method can lead to accumulating positional errors, causing the end-effector to deviate from the desired path. This error arises from the control discretization and physical limitations on the motor acceleration.

Increasing the control frequency or using more powerful motors will reduce the error but cannot eliminate it. To address this, we develop the Implicit Euler Velocity Control (IEVC) algorithm, discussed in Section 5.1.2.

For

individual users. Previous research and commercial products often focus on predefined trajectories with symbolic solutions (7, 9, 10). While assistive force calculations for symbolic

Complex,

customized trajectories requires additional fitting and tuning. We propose a novel control method that efficiently calculates assistive forces on a 2D discretized numerical trajectory. Unlike symbolic methods, this method handles any trajectory in a 2D workspace without fitting and tuning. With computational time

Uniquely

supports real-time customization.

2 Related Works

End-effector upper limb rehabilitation devices refer to planar robots enabling movement along a 2D plane. Numerous studies have investigated rehabilitation techniques using these 2D planar robots through rigorous experimental methodologies (6–8, 11–14).

There are three primary types of end-effector rehabilitation robots: multi-link robots (6, 11–13), pulley-driven robots (7, 14), and linear actuated robots (8) (Figure 1). The MIT MANUS is a well- known example of a multi-link rotary robot with a two-decade history (6). Stability of the vertical force, resulting from the user placing their arm on the device, has been a notable concern for multi-link rotary robots. Despite their portable and backdrivable direct drive mechanisms providing high accuracy, maintaining stability of the vertical force remains challenging (7). Researchers proposed pulley-driven robots and linear actuated robots to improve stability (15, 16). Pulley-drive robots attempted to employ a differential drive with pulleys and belts, to address the stability issues and improve dynamics calculation (7, 10, 14, 17). However, the pulley-drive robots require high maintenance due to the frequent wear and tear on the belt drives. To our knowledge, only

Rehabilitation

devices.

Arm

rehabilitation device, commercialized based on the multi-link MIT

(6)

design.

Arm Rehabilitation Device,

utilizing an H-shaped differential pulley and belt mechanism. (C) Physiobot arm rehabilitation device using a backdrivable linear actuated mechanism (8). (D) ArmMotus M2 Gen (9) rehabilitation device utilizing a backdrivable linear actuated mechanism. (E) Our Gantry Arm Rehabilitation Device (GARD) operates in Assist-As- Needed Mode along a circular trajectory. The GARD can provide either assistance or resistance to support physical therapy training.

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one commercialized device claimed to solve these reported issues

With

minimum information revealed (8, 9). The current paper explores

Non-Backdrivable

mechanism to provide less maintenance than pulley-driven robots. End-effector upper limb rehabilitation devices typically offer

And

Transparent Mode. These training modes aim to map the sequential stages of conventional physical therapy progression (14, 18, 19). The Powered Mode is primarily utilized in the early stages of rehabilitation for stretching and passive range of motion (ROM) exercises (20). Patients follow a predetermined 2D trajectory operated by the device, without using strength or exerting forces. The Assist-As-Needed Mode attempts to facilitate active range of motion (ROM) exercises and resistance training (20, 21). In this mode, the device provides assistance or

As

needed.

Users

unrestricted movement within the 2D workspace, facilitating engagement with rehabilitation games, promoting sensorimotor training, and enhancing tactile feedback, motor planning, and sensory and proprioceptive awareness (22).

3.1 Mechanical Design

The Gantry Arm Rehabilitation Device (GARD) end-effector offers a flexible range of motion within a 65
55 cm 2D workspace. This device utilizes two rotary actuators coupled with ball screws, enabling linear movement along the x and y axis, as depicted in Figure 2. The ball screws operate at a ratio of 1 turn to 2 mm. The actuators can achieve a maximum speed of 80

800

revolutions per second squared. Additionally, each actuator provides a maximum torque output of 0.22 Nm.

3.2 Electrical Design

The device uses a medical-grade power supply with a rated output of 24 V and 6.5 A, distributed via a power board to components such as the motor controller, force sensor, and controller board. Two motor controllers are connected to Brushless DC motors with 1000CPR optical encoders, enabling precise control of the x and y-axis ball screws. Additionally, the system employs two limit switches for motor-homing purposes.

For force sensing at the end-effector, a low-cost Galoce GPB160 3-axis load cell is employed, utilizing an ADS1232 SPI ADC converter. To manage low-level control and communication

Adc

converters, and motor drivers are all interfaced with an STM32 microcontroller unit (MCU).

3.3 Control Hierarchy

GARD uses a two-level control hierarchy consisting of a High- level Virtual Dynamic Model and a Low-level Firmware Controller, as depicted in Figure 3A. The High-level Virtual Dynamic Model calculates the desired motor velocity from sensor data and user settings.

Data

collection, timing, and motor control.

Figure 2

Mechanical design of GARD. (A) The top view illustrates the 65
55 cm 2D workspace attained through the use of two rotary actuators and ball screws. (B) The front view showcases the y-axis actuator and ball screw, which can move both the end-effector and the x-axis actuator with the ball screw in the y-direction. The x-axis actuator moves the end-effector in the x-direction.

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The current structural setup diverges from conventional admittance control systems. In conventional systems, the virtual dynamics model and motor control cannot be separated due to the influence of applied forces on robot dynamics. However, the GARD features a non-backdrivable end-effector, where user force does not influence end-effector motion, allowing the separation of the virtual dynamics model and the motor control. The

Of

commercial motor drivers. The integrated commercial motor driver firmware is more stable and efficient than the user’s implementation on a multitask MCU because they are dedicated to motor velocity control.

4 Gard Operation Modes

Three distinct operation modes have been implemented to support patients at different stages of recovery: Powered Mode, Transparent Mode, and Assist-As-Needed Mode. A Graphical User Interface (GUI) is provided for all operational modes, offering GUI functionality and rehabilitation gaming features.

Figure 3

(A) The overall control hierarchy of GARD. A high-level Virtual Dynamics Model computes the desired motor velocity and sends it to the Low-level Firmware Controller, which manages sensor data collection, timing regulation, and motor control. Fsmp denotes data from the force sensor. Pin denotes the current position of the end-effector. Yv denotes the virtual dynamics. vd denotes the desired velocity. Cm denotes the motor controller. Ym denotes the robot dynamics. (B) Control block diagram for Powered Mode. (C) Control block diagram for Transparent Mode and Hard Boundary Assist-Ad-Needed Mode (D) Control block diagram for Soft Boundary Assist-Ad-Needed Mode. Details for modes and control blocks can be found in Sections 4, 5.

Li et al.

Their

corresponding control algorithms.

4.1 Powered Mode

Powered Mode is designed for passive rehabilitation, where the robot follows a pre-set trajectory at a user-defined speed. In this mode, exerted force on the end-effector does not influence the robot’s motion. It is typically used during the early stages of recovery when external guidance is needed to complete arm movements.

The Powered Mode control hierarchy is shown in Figure 3B. The GUI controller collects user settings and inputs, sending them to the Motion Restriction Map Generator and the Velocity Regulator. The Velocity Regulator outputs a velocity vector based on the desired speed and current velocity direction. The Motion Restriction Controller adjusts the velocity vector to keep the end- effector within the desired moving area. We propose the IEVC algorithm that implements the Motion Restriction Controller function. The Motion Restriction Controller and IEVC are reused in all modes. More details are provided in Section 5.1.

4.2 Transparent Mode

Transparent Mode is an interactive mode used in the later

And

sensorimotor training. The Transparent Mode allows the user to move the end-effector within the user’s Range of Motion (ROM). ROM is the area where users can safely move their arms without injuries. The interaction with the ROM boundary resembles encountering a hard-smooth-wall-like boundary.

Figure 3C illustrates the control hierarchy for Transparent Mode. The Admittance Virtual Dynamics block simulates the behavior of a mass on a rough surface, with friction and damping effects. Further details about this system can be found in Section 5.2.

4.3 Assist-As-Needed Modes

Assist-As-Needed Modes are used for both active Range of

Resistance

training.

The

Transparent Mode, this mode allows free movement of the end- effector within the training area. However, the Assist-As-Needed Mode focuses on trajectory-based training, whereas Transparent Mode focuses on area-based training. There are two versions of this mode: hard boundary and soft boundary.

In the Assist-As-Needed Mode with hard boundary, the user can move the end-effector freely along a fixed trajectory. The user is restricted from deviating off the path similar to following a rail. The control structure of this mode is identical to the Transparent Mode. This is because a trajectory restriction and an area restriction are identical under our proposed algorithm.

The soft boundary version of Assist-As-Needed Mode is similar to the hard boundary version but allows the user to deviate from the path. The device applies automatic assistance force to guide the user back to the trajectory. The control hierarchy of the Assist-As-Needed Mode with soft boundary is

Block

calculates the assistance force for trajectory realignment. Details of the Impedance Controller can be found in Section 5.3.

5 Control Algorithms

This section covers the control algorithms that enable GARD’s operation modes. Section 5.1 explains the motion restriction and trajectory-following algorithms, Section 5.2 covers the simulated virtual dynamics, and Section 5.3 introduces the automatic assistance force calculation algorithms for soft boundary Assist- As-Needed Mode.

5.1 Motion Restriction Algorithms

Truncation error accumulation: A common practice in implementing the trajectory-following algorithm is to decompose the current velocity into tangential and radial velocity and adjust them accordingly. However, these types of algorithms share a common issue: the position error can accumulate over time, resulting in increased deviation and undefined positions. Take circular trajectory as an example: a trivial way to follow a circular trace is to dynamically calculate the tangential direction at the current position and march only in that direction.

However, with this method in practice, the error caused by the control discretization and the motor acceleration limitations will accumulate, and will eventually result in an outward spiral trace instead of a circular trace. This example of truncation error

Error

accumulation in solving the initial value problems using the Explicit Euler Integral (23). We propose the Implicit Euler velocity control (IEVC) algorithm to address the error accumulation issue during free moving along a trajectory restriction. This algorithm predicts the end-effector’s next position and adjusts the current velocity, inspired by the Implicit Euler Method (23). However, we discover that this method can solve more than trajectory restriction: it can be naturally extended to 2D area restriction or even 3D spatial restriction. Hence, we define the Motion Restriction Control as the end-effector moving control under a virtual positional restriction. Areas that the end-effector is allowed to move are the permitted area, otherwise are the prohibited area. A Motion Restriction Map is a 2D matrix that

And

prohibited areas. The proposed IEVC algorithm can effectively restrict the end-effector inside any permitted area, used in Figure 3 as the Motion Restriction Controller block. The IEVC allows the end- effector to move freely inside the permitted area and interact Li et al.

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with a hard-smooth-wall-like area boundary. The following two subsections discuss the details of the Motion Restriction Map and IEVC algorithm.

5.1.1 Motion Restriction Map Generation

The task space of GARD is discretized into Wg
Hg positions, forming a 2D matrix of equal size referred to as the Motion Restriction Map. The Mrs can be visualized as a grayscale image.

Each pixel entry on the Motion Restriction Map corresponds to a physical GARD end-effector location. A pixel storing a value of 0 indicates that the corresponding location is the prohibited area; Any non-zero value indicates the corresponding location is the permitted area.

Will Be

initialized with all 0, meaning all prohibited areas. For a trajectory restriction defined using implicit functions f (x, y) ¼ 0 or more generalized area restriction defined using F(x, y) , 0, the Motion Restriction Map can be generated by iterating through all entries [i, j] and evaluating abs(f (j, i)) , Es or F(j, i) , 0. Es is the trajectory width for trajectory restriction.

If position [i, j] meets the condition, the matrix entry will be assigned 1, marking it as a permitted area. For other general user-defined motion restrictions such as a hand-drawn trajectory restriction, Mrs can be generated or edited using image editing tools.

5.1.2 Motion restriction controller (IEVC algorithm) In simple terms, the IEVC algorithm restricts motion within the permitted area by predicting all possible positions for the next time step and selecting the optimal one. The optimal position will be set as the chasing target. There are three

Requirements For The Chasing Target:

1. The chasing target needs to be at a suitable distance according to the current speed. 2. The chasing target needs to be inside the permitted area.

3. The chasing target needs to be as aligned with the current velocity direction as possible. The first requirement accounts for the fact that a higher end- effector speed means it will be farther in the next time step. The second requirement ensures the end-effector is always directed toward the permitted area. The third requirement filters the best candidate and allows the algorithm to be generalized into the area-based motion restriction.

Figure 4 illustrates an example of IEVC output calculation. The algorithm will first draw a circle centered at the current position Pin with a radius proportional to the current speed magnitude (r ¼ GvkVink). All positions on this circle satisfy requirement 1. The proportional gain Gv is calculated by the transmission ratio between the motor and the end-effector. The algorithm then identifies intersection points between the circle and the permitted area, marking them as candidates. In the figure, 2 candidate points are marked with blue dots, both of them satisfy

And

2.

All

candidates and chooses the one with the most aligned velocity as the chasing target. This target will satisfy all three requirements. Finally, the algorithm generates the final output velocity Vr using the chasing target direction and the projection of current velocity

The

magnitude.

The

momentum loss on collision with the wall, creating a hard-

Area

boundary.

The

example in Figure 4 also demonstrates that even when Pin is outside the permitted area (trajectory), the error will not accumulate under IEVC. The divergence distance will decay exponentially over time steps.

Algorithm 1 present the sudo code for IEVC. Pin denotes current position, Vin denotes current velocity, Mrs denotes the motion restriction map, Gv denotes the transmission gain. The helper function sudo codes can be found in Algorithm 4.

5.2 Admittance Virtual Dynamics

The Admittance Virtual Dynamics block is used in both Transparent Mode and Assist-As-Needed Mode. This control

Dynamics,

allowing the natural motion of the end-effector. Given a simulated mass, we have the following equation of

(1)

where, mv denotes the virtual mass. kr denotes the transmission ratio between the motor and the end-effector velocity. Ts denotes the time step size.

The Laplacian domain model cannot be directly used for discrete control. There are two steps to convert the Laplacian domain model into discrete control. First, take the Z-transform

(2)

Second, take the inverse Z-transform on Equation 2 gives the time domain equation which can be used for discrete control:

(3)

Algorithm 2 shows the implementation of Equation 3. The algorithm takes current input force Fin, input force on last time step Flast, velocity on last time step Vlast, virtual mass mv, damping ratio za, and friction coefficient m as input, and will output the desired velocity Vd. The Vd will be sent to the Motion Restriction Controller. The motor velocity from the Motion Restriction Controller will be sent to low-level GARD

Close-Loop

velocity control. Li et al.

5.3 Impedance Controller

The Impedance controller calculates assistive forces based on the current position and the motion restriction map. It operates in conjunction with the Admittance Virtual Dynamics in the soft boundary Assist-As-Needed Mode, simulating a spring-damper (impedance) boundary for the permitted area.

The output from the Impedance Controller is a composite force vector that combines: the end-effector sensor force, the simulated impedance spring force (referred to as spring force hereafter), and the simulated impedance damper force (referred to as damper force hereafter).

We propose a novel algorithm that efficiently calculates the impedance force on a Motion Restriction Map in real-time. This method provides a general solution for the discretized workspace, meaning that the complexity of the Motion Restriction Map has no impact on the computational cost. The algorithm works by precalculating a special map called the Spring Force Map using 2D convolution. The Spring Force Map allows the efficient calculation of impedance force at any given position in O(1) time.

Example

To explain the need for convolution, consider a simplified impedance force calculation problem in 1D. As illustrated in Figure 5A and Figure 5B, Mrs(x) is our example 1D Motion Restriction Map. Each x value corresponds to an end-effector location on the x axis, meaning all positions between 2  x  6 are permitted area. Kernalsw(x) is a unit impulse function and is used as the convolution kernel. The convolution result, seen in

Figure 4

Illustration for IEVC calculation at position Pin with velocity Vin. In this example, the IEVC finds 2 candidates according to the intersection circle centered at Pin with radius GvkVink. Candidate 1 is selected due to higher directional alignment with Vin. The output Vr is Pin’s projection onto candidate 1’s direction. The IEVC is stable because the restricted velocity given by IEVC always move the end-effector towards permitted area. In the example above, even when current position Pin does not lie inside the permitted area (trajectory), the error will decade exponentially over time steps and IEVC can still effectively restrict the end-effector location on the area boundary.

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