1. Introduction
Life expectancy continues to increase, accompanied by motor disabilities in the elderly population, and exercising is an important way to prevent their depletion [
1,
2]. Particularly, neurological and orthopedic lesions can affect upper limb mobility [
3]. Therefore, the restoration of the ordinary function of the upper limb is essential through therapy exercises [
4].
The aim of the rehabilitation robotics is to assist and support medical activity during a therapy as well as to accelerate the patient recovery process and to maintain the human health. An early recovery of sick or injured people is important for their integration into daily life. Therefore, since patients require assistance during the execution of repetitive movements and robotic system advantage this process [
5,
6], a lot of devices have been developed to support the upper limb exercises. For example, in [
7] a support system is projected to allow movement on vertical elbow flexion and shoulder/elbow horizontal flexion and extension. However, the device only conducts fundamental motions, but it does not carry out other paths. For the upper limb exercises another mobile device is suggested in [
8]. The device is comprised of an H-shaped cable-driven machine, two motors, and a hand grip. The system has four guides that restrict the motion of the end-effector in a vertical line, a horizontal line and two 45-degree diagonal rows. The path movements include the shoulder and the elbow. Nevertheless, since mechanism paths are restricted by four guides that limit the end effector along right lines, it is impossible for the mechanism to do other exercises. Moreover, without the flexibility required to adapt the practice/therapy to individual needs all patients should carry out the same trajectories. In addition, the patient response can modify the tension of the wires because the unit has a cable-driven system. Another portable device is presented in [
9]. The device is composed by a base plate, hydraulic damper, restrictor arm, actuator arm, elbow cup, and hand grip. The device can assist the internal and external rotations of both right and left shoulders. Using the device, the arm should be attached to the device so that the elbow is located in a fixed position and the hand should grasp a handle. The device can get a horizontal or inclined position. However, the device can only perform one exercise and it can only assist the shoulder joint but not the elbow.
Although existing systems have a large workspace, they are very difficult to carry, build, and wear. In [
10], the authors proposed a planar three DoF (degrees of freedom) exoskeleton robot that assists horizontal motion for shoulder, elbow, and wrist. The exoskeleton is controlled by a cable-drive system. The user’s arm is located on the top of the robot structure. The distances between the axes can be adjusted for each subject. Nevertheless, since the apparatus requires a voluminous framework structure, transportation and construction is hard. The biomedical robot is furthermore adjustable only for the right upper limb, and the design has problems in aligning the mechanism joints with the upper limb joints. An exoskeleton powered by a cable is presented in [
11]. However, the exoskeleton is hard to dress because its exoskeleton structure surrounds the upper limb. However, it offers additional exercises regarding the exoskeleton in [
10]. Another example is presented in [
12], where a biomedical device is proposed for a robotized rehabilitation for a human upper limb. The biomedical device is an exoskeleton type that is principally composed by two rigid rods that are associable with the forearm and the upper arm by joints with four degrees of freedom. This device has a bulky frame structure that is difficult to transport. In addition, the exoskeleton structure is hard to wear.
Commercial devices are currently also available for upper limb exercises. In [
13,
14], a commercial exoskeleton with ergonomically actuated shoulder is presented. However, it has voluminous bars and frame. In addition, it is hard to wear and it is expensive. In [
15,
16], a popular device that consists of a visual interface and a robotic arm to guide the patient’s arm along desired paths is presented. A five-bar mechanism is used to drive the robotic arm by two motors, which execute horizontal motions. In addition, a modular end-effector assists the wrist joint movements. However, the device is very complex and difficult to operate and requires well-trained staff to guarantee safe operations. Moreover, the device is expensive, heavyweight and it has a very tiny workspace for the exercises. In [
17], a sophisticated commercial device based on the exoskeleton in [
13,
14] as described in [
18] is presented. The exoskeleton has six degrees of freedom to perform 3D motions and a graphical interface for virtual interaction. However, this exoskeleton presents the similar disadvantages of the exoskeleton in [
13,
14]—a bulky frame, difficulty to wear, and costly. ReoGo is a commercial and portable robotic arm [
19] that has a mechanism similar to a joystick. ReoGo performs two-or three-dimensional movements and it also has an interface for virtual interaction. The disadvantage of ReoGo is its small workspace. NeReBot is another device for arm therapy [
20]. NeRebot is a cable-suspended device of three DoF with three cables whose end-effector moves inside a spatial working space. The robot cables are driven by three electric motors. The cables are connected to the patient’s upper limb by using an arm support whose end-effector moves inside a spatial working space. The structure to support the cables can be adjusted manually. The NeReBot performs repetitive passive movements of the upper limb and it can be used by having the user in a chair or a bed. However, NeRebot is heavy and presents some disadvantages [
21]—it is bulky and some horizontal movements of the upper limb cannot be performed properly.
Rehabilitation centers often use the skateboard instrument to assist the training exercises for the upper limb [
22]. The skateboard consists of a wheel board that enables horizontal movement. Using the skateboard, a therapist guides the movement of the patient’s arm and the patient does the movements by himself. The skateboard is an affordable system, but usually does not have a regulated movement. In [
23], a portable device for arm therapy similar to a skateboard is proposed. The mechanism consists of a mobile platform with three spherical wheels. The platform is attached to the forearm and it can perform movements in a horizontal plane. The device has an optical tracking system that is composed of two optical mouse sensors. However, since the platform is not attached to a fixed frame the patient’s reaction can easily alter the reading of the mechanism positions.
As described in the above paragraphs, the existing devices has several issues.
Table 1 shows a summary of advantageous and disadvantageous features of the referenced devices. Regarding exoskeletons and semi-exoskeletons in [
10,
11,
12,
13,
14,
17,
18], despite offering a large workspace they present bulky frames that hinder its transport and its building. Most of them use of bulky links that must be moved by the motors requiring greater torque than if light links. In addition, the exoskeletons are difficult to wear since they should surround the patient’s arm as if it were a sleeve. Moreover, the joint axes of exoskeletons should be aligned with the anatomical axes of the human arm and it is hard to get it considering that the human arm sizes are different for each subject. In addition, it should be mechanically reconfigured for right and left human arm. On the other hand, the portable devices referred in [
7,
8,
9,
19] are portable devices with light frames which could eventually be used as home devices with the proper supervision of a specialist. However, they have predefined paths that cannot be modified due their mechanical shapes, thus the number of exercises or trajectories that they can offer is very limited. Although manipulator devices in [
15,
16] can perform several paths, they work with a limited workspace considering the human arm workspace on the horizontal plane and the mechanism sizes. In [
23], the device is a controlled skateboard design that can be moved as the patient requires. However, since it is not attached to a fixed frame the patient’s reaction can easily alter the reading of the mechanism positions. Moreover, the device in [
23] is a patented idea and practical designs or experiences are not available to consider its feasibility. On the other hand, the proposed solution in this work has aimed to design a mechanism that merges together most of the advantageous features of existing mechanisms but avoids the disadvantageous features that each one of them presents. The proposed device has been designed to offer a large workspace without bulky frames to guide both the left and right upper limbs without the need to be mechanically reconfigured, to follow desired paths by programming the device avoiding that the mechanical structure predetermine paths that cannot be modified by programming, to offer a light structure for portability and therefore facilitate rehabilitation at home with the proper tele-supervision of a specialist, to provide a design with light links reducing the load that the motors of the mechanism must move reducing the required torque and consequently the power consumption that is directly proportional, and to offer the possibility of programming different paths customized for people of different sizes and needs such as children and the elderly people without the need of mechanical reconfigurations or the need of a kid’s version design.
2. Mechanism Task
Medical experts at the Center of Rehabilitation of Queretaro in Mexico (CRIQ) reported that multiple training sessions during arm treatment are usually performed on a table where the patient’s arm is supported by a skateboard [
22,
24]. The exercises consist of performing horizontal movements on a suitable table while the skateboard supports the patient’s arm and the therapist assists the arm motion. However, existing devices that assist the arm motion on horizontal plane, as reported in the introduction section, have several issues that should be solved. Therefore, an innovative mechanism has been required to guide the human arm motion on a horizontal plane with advantages over the existing devices. The mechanism task is to guide the movements of the human arm along predefined paths on a horizontal plane.
A manipulator has been considered as an appropriate mechanism type to assist the arm exercises on horizontal plane since the patient’s hand could be assisted by the mechanism end-effector similar to when the therapist does it. So that, the mechanism end-effector should perform movements along
X and
Y coordinates and it should reach most of the workspace essential to perform upper limb exercises on a horizontal plane. The space needed for horizontal arm training has been estimated using mean anthropometric parameters for the upper limb lengths [
25,
26] (See
Figure 1). The average length of the arm, from the shoulder to the middle of the hand, has been calculated as 667.5 mm. Consequently, the space required for horizontal arm training has a maximum height of 667.5 mm and a maximum width of 1335.0 mm. On the other hand, multiple exercises can be achieved by the upper limb on a horizontal plane as described in [
22,
24]. However, in order to test the proposed device, one exercise has been chosen. To carry out the exercise, the number eight path should be traced by the manipulator end-effector. The coordinated motions to carry out the number eight path is complex enough to test the proposed device [
24]. X and Y references versus time to perform the exercise has been obtained from previous experiments whose results have been reported in [
24,
27] where arm exercises on horizontal plane were designed using several samples of healthy individuals and processing the references by regression analysis.
Figure 2 shows
X and
Y cartesian coordinates to trace the number eight.
3. Kinematic Analysis and Design Parameters
To control an end-effector position consisting of cartesian compounds
X and
Y, two DoFs are needed. On the other hand, a planar mechanism is enough to perform movements on the planar workspace in
Figure 1. Therefore, a planar mechanism of two DoFs has been proposed (See
Figure 3). The suggested mechanism consists of by a five-bar and a pantograph mechanism. The pantograph was designed to enlarge the five-bar workspace and therefore to use tiny connections. The five-bar mechanism mobile bars are
L1,
L2,
L3, and
L4 and the pantograph mobile bars are
L6,
L7,
L8, and
L9. The bar
L5 is the attachment structure set frame. Point
E is the pantograph tracing point which is associated to the five-bar mechanism end-effector. Point
F is the amplified tracing point to follow the arm exercise path.
M1 and
M2 are the active joints that will be actuated by motors and they are related with
and
angles, respectively. The distance
H is defined as
H =
L1 =
L2 =
L3 =
L4. It is important to notice that variable
H is included in the kinematics formulation. The
,
, and
angles are required for the kinematic analysis. The point
A,
E, and
F are always aligned for any position of the pantograph so that a straight line connects the three points. The proposed mechanism has been protected by the patent in [
28].
The rotational movements of
M1 and
M2 were reduced from 0° to 180° to prevent physical collisions. In addition, in order to obtain the maximum possible workspace using the five-bar mechanism [
29,
30,
31] the bars
L1,
L2,
L3, and
L4 have equal lengths so that
L1=
L2=
L3= L4, and the joint
M1 is aligned with joint
M2. Distance |AC|/|AB| give the amplification scale of the pantograph. The dimensional ratios
|BE|=|CD|;
|BC|=|ED|;
|AB|=|BE|; and
|AC|=|CF| define the pantograph structure [
32]. The mechanism end-effector should, according with the workspace in
Figure 1, cover a maximum height of 720 mm and a maximum width of 1440 mm based on an about 5 cm safety factor. From
Figure 3, the angle
can be defined as:
where
YF and
XF are the coordinates of the point
F (end-effector positions) with respect to the Cartesian reference frame.
YF and
XF are known inputs in the inverse kinematics. In
Figure 3, the triangle formed by points
A,
C, and
F can be defined by using law of cosines in the form:
On the other hand, by using the Pythagorean theorem,
can be defined as:
then, replacing Equation (2) in Equation (1):
So that the angle
β1 can be computed from Equation (4) in the form:
where
is the angle between links
L6 and
L9. In a pantograph mechanism
is defined as:
In addition, the triangle formed by points
A,
B, and
E can be identify by using law of cosines in the form:
where
|AB| is the distance between the point
A and the point
B. The variable
can be isolated from Equation (7) as:
Then, the angle
in Equation (8) can be substituted by
from Equation (6). The angle
is a known variable since it can be calculated using Equation (5). Therefore, Equation (8) can be rewritten as:
Since the angle
and the variable
can be respectively calculated using Equations (1) and (9), the Cartesian coordinates of the point
E with respect to the Cartesian reference frame can be defined for both solutions in the form:
Using
and
values, the angular positions
and
of the active joints
M1 and
M2 can be computed for both solutions as follow:
where:
and:
where:
The inverse kinematics is formulated by Equations (12) and (16). Using the inverse kinematics equations, the mechanism workspace can be determined. However, first the link lengths must be defined so that the end-effector can cover most of the workspace of the human arm motion in
Figure 1. Therefore, the limits of the mechanism workspace have been approximated by indicating a maximum height (in
X-axis) of 720 mm and a maximum width of 1440 mm (in
Y-axis) when a safety factor of approximately 5 cm is considered [
24].
According to the limit of movement of M1 and M2 (0° to 180°), and the relationship between the bars (L1 = L2 = L3 = L4), five-bar can reach a maximum distance equivalent to the length of each symmetrical link L1, L2, L3, and L4. Thus, the link lengths of the five-bar should be equal to 720 mm to cover the needed workspace without taking in consideration the pantograph. Therefore, since the pantograph can amplify the workspace of the five-bar, a decreases scale of 1/4 has been implemented to the link lengths of five-bar, obtaining that L1 = L2= L3= L4 = (720 mm/4) = 180 mm.
Consequently, the pantograph needs to increase the five-bar workspace by a factor of 4 to cover the required workspace. Therefore, the amplification scale of the pantograph should be |AC|/|AB| = 4. Since the angles and have been delimited from 0° to 180°, the distances |AB|+|BE| should be equal to 180 mm so that the tracing point E reaches the usable workspace of the five-bar. According to the dimensional ratios of the pantograph, since |AB|=|BE| and |AB|+|BE| = 180mm, then |AB|=|BE| = 180 mm/2 = 90 mm; since |BE|=|CD|, then |CD| = 90 mm; since the amplification scales is defined by |AC|/|AB| = 4, then |AC| = 4 × 90 mm = 360 mm; since |AC|=|CF|, then |CF| = 360 mm; since |BC|=|AC| – |AB|, then |BC| = 360 mm – 90 mm = 270 mm; and since |BC| = |ED|, then |ED|= 270 mm. Subsequently, L6 = 360 mm, L7 = 270 mm, L8 = 90 mm, and L9 = 360 mm.
Since the link sizes were defined, the workspace can be calculated using Equations (12) and (16).
Figure 4 shows the mechanism workspace (in black) versus the workspace of the upper limb motion on a horizontal plane (in grey). The circles in
Figure 4 indicate the uncovered areas of the workspace of upper limb motion. It is important to notice that the patient should be located in a convenient and comfortable place in front to the end-effector (point
F in
Figure 3) at the opposite side of the active joints. The uncovered areas of the workspace of the upper limb motion are not necessary to perform the proposed exercise and the exercises reported in [
24,
27].
The calculated workspace shown that the proposed mechanism design is able to cover most of the workspace of the human arm on horizontal plane using the proposed link sizes. Consequently, it proved the proposed mechanism need link sizes of maximum 360 mm to cover the required workspace allowing a compact design. In addition, it shown that arm exercises reported in [
24,
27] can be reproduced within the mechanism workspace since it is large enough.
4. Dynamic Analysis
The feasibility of the suggested linkage mechanism was evaluated by a dynamic analysis using ADAMS solver within SolidWoks software (version 2018, Dassault Systèmes, S.A., Suresnes, Francia) simulating the mechanism performs the exercise path showed in
Figure 2 in
Section 2. In the simulation setting, a 1060 aluminium alloy has been used for the mechanism links.
Table 2 shows the assumed characteristics of the chosen material. The parameters of friction and gravity force are shown in
Table 3. The simulation considers an integrator step size of 1.0 × 10
-4, with a maximum value of 1.0 × 10
-1 and a minimum value of 1.0 × 10
-11, 800 frames per seconds and an accuracy of 1.0 × 10
-9. A time of 8 seconds is required to perform the selected exercise path as reported in [
24]. Considering the average weight of the upper limb [
25,
33], a force of 37 N has been applied in the amplified tracing point of the mechanism (Point
F in
Figure 3 in
Section 3) perpendicular to the working plane (Z axis). It is important to note that the amplified tracing point of the mechanism (Point
F in
Figure 3 in
Section 3) must track the desired exercise path showed in
Figure 2,
Section 2.
It is important to note that in this dynamic analysis the motors of the mechanism are simulated using the trend of velocities required to perform the trajectory to trace the number 8 as the input parameter, see
Figure 5a. The dynamic analysis outcomes are the torque, angular displacement, and angular acceleration required by the simulated motors to perform the prescribed trajectory. Angular displacement, angular acceleration and torque obtained from the dynamic analysis are shown in
Figure 5b–d, respectively.
Table 4 shows the results for motor 1 obtained from the dynamic analysis as well as the maximum and minimum velocity used as input for motor 1.
Table 5 shows the results for motor 2 obtained from the dynamic analysis as well as the maximum and minimum velocity used as input for motor 2. The dynamic simulation outcomes allow to choose the motors for the mechanism construction and to check the efficient operation.
Referring to
Figure 5, and the numerical values reported in
Table 4 and
Table 5, the angular displacements present a symmetric mechanism performance and the computed accelerations and torques values confirm the possibility to use inexpensive commercial motors. The proposed mechanism has a satisfactory behaviour in achieving the desired objective since it presents a smooth motion and limited torque (referring to plots in
Figure 5). Therefore, according to the design and operations requirements, the proposed mechanism can be considered a feasible design to assist the human arm exercises. The mechanism can perform successfully the exercise to trace the number 8 with displacement, acceleration, velocity and torque achievable by commercial motors, referring to
Figure 5. In addition, the required torque, shown in
Figure 5d, allows a low power consumption to achieve the desired task. Even though the mechanism has been tested via simulation with a specific exercise, other exercises can be also performed by the mechanism within its workspace.
Since the assistive mechanism can perform trajectories of different sizes and shapes within its workspace, the exercises can be customized for people of any anthropometric sizes, including children and elderly people. In addition, the exercises can be customized for physical therapy, for treatments of injuries or diseases, for prevention, or for physical exercising.
5. FEM Analysis
Since the proposed mechanism should assist arm exercises that are normally performed over a work table, proper commercial spherical wheels have been added in the mechanical design in order to give stiffness to the linkage structure and use a work table as support (See
Figure 6). Furthermore, an ergonomic interface has been studied and added to the structure to allow the patient to lean the arm and distribute its weight. The mechanism structure has also been attached to a fixed frame with “T” shape for a portable design that can be installed on a table or a flat plate. The fixed frame allows also to use flat motor or other rotational motors.
A linear static FEM (Finite Element Method) analysis has been performed to test the feasibility of the prototype and to check the response of the proposed structure to the considered applied forces. All the device components have been considered as made of aluminium 1060 alloy with linear elastic isotropic properties as in
Table 2. A soil has been considered during the FEM analysis in order to simulate the work table support for mechanism operation. Gravity force has been considered acting along the Z axis. The average weight of the arm has also been assumed as 37 N for the force acting on the mechanism end-effector during the simulation considering the average weight of the upper limb [
25,
33]. The horizontal forces that can come from the human arm have been considered negligible for the current static analysis. However, they can be considered in a future work using a dynamic FEM analysis. To analyse the model using the proper criteria a mesh is needed.
Table 6 shows the mesh information.
As Stress Analysis criterion, Von Mises stress function has been used as a measure of all stress components of a general 3-D state of stress. Von Mises stress function σ
vm, can be expressed by stress components in the form [
34]:
where σ
1, σ
2, and σ
3 are the three principal stresses acting on X, Y, and Z-axes of the body. Von Mises stress is a non-negative scalar stress measure that evaluates elasto-plastic properties. This number function represents a stress magnitude, which can be compared against the yield strength of the material in order to determine whether or not failure by yielding is predicted.
Using the maximum Von Mises stress criteria, [
34], the condition of safe design can be expressed as:
If the Factor Of Safety (FOS) is 0 we are in a critical condition and if it is <1 the material failed. Displacement along Z, stress and FOS of the proposed mechanism have been obtained from the FEM analysis.
Figure 7a shows the mechanism displacement along Z axis that has a minimum value –1.60 mm and a maximum value of 0.73 mm. Therefore, the reached displacement along Z can be considered negligible.
Figure 7b shows the computed stress that has a range between 6.39 × 10
-7 N/m
2 and 4.45 × 10
6 N/m
2. Therefore, the stress is very far from the yield limit of the material reported in
Table 2. The stress results show that the structure is stiff and it can support the applied forces using light bars.
Figure 7c shows the computed Factor Of Safety (FOS) check. The FOS values reach a minimum of 6.19 that can be seen in the red zones. Finally, the FOS check shows that no nodes reach failure since its minimum value is 6.19.