Full or partial loss of function in the shoulder, elbow, wrist or hand is commonly known as the physical disability of the upper limb motion. Strokes, sports injuries, trauma, occupational injuries, and spinal cord injuries are some reasons standing behind this malfunction [1, 2].
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Loss of upper limb motion causes loss of essential daily living activities. Fortunately, there are various approaches to restoring the functionality of the upper extremity, like surgery, replacement for the detective organs, functional electrical stimulation, and physical therapy.
Rehabilitation robotics, either end-effector-based or exoskeleton, is a tool for clinicians in rehabilitation programs instead of traditional ones [3]. They are used for their ability to provide accurate and repeatable movements over long durations. Robotic devices can also be leveraged to record performance data for tracking the therapeutic progress of patients [4–6].
End-effect-based robotics are devices that do not actively support or hold the subject’s upper-limb, but connect with the subject’s hand or forearm. An end-effector rehabilitation robot is one which interacts with the user only with the end-effector. This type typically allows for large functional workspaces but does not mirror human anatomy and is thus unable to apply torques directly to human joints [7–12].
Exoskeletons, on the other hand, are anthropomorphically designed where robot joint axes are typically collocated with human joints axes. They allow for the direct application of torque to individual joints. This mapping between robot and human movement makes exoskeletons more attractive than end-effector designs for rehabilitation robotics [13].
Exoskeletons are the majority of robots used for rehabilitation processes. They are classified according to the actuators as rigid or soft robots.
D. Buongiorno et al. [14] have developed a rehabilitation robot, WRES, with 3-DOFs for the forearm and wrist joints, driven by brushless DC gear motors and used differential transmission for joint movements. Another robot used for rehabilitation of forearm and wrist joints with 3-DOFs was developed by Y. Su et al. [15]. Stepper motors were the actuators of this serial kinematics-based robot. Tanvir Ahmed et al. [16] had developed a 3-DOFs rehabilitation robot for the forearm and wrist joints. It was comprised of three joints responsible for providing three different movements to the human forearm and wrist using brushless DC motors attached to aluminum links.
Rigid robots have some disadvantages compared with soft ones such as high weight, high cost, low power-to-weight ratio, higher risk of injury and high maintenance processes. Also, these robots cannot be adjusted for each person’s demand because of the variety of patients’ limbs. Soft actuators overcome these limitations. They are inherently comfortable and safe for direct human interaction because of their light weight and lack of rigid parts.
Soft actuators can be classified as actuators with low stiffness materials or pneumatic actuators like pneumatic artificial muscles (PAMs). However, low stiffness materials-based robots are still under research due to their limitations in terms of not covering both the range of motion and the required torque at the wrist joint. PAMs based robots are more advantageous because of its high power-to-weight ratio, fast and accurate response, lightweight, high strength, and no requirement for mechanical parts [17, 18]. Unfortunately, they are difficult to control due to their nonlinear characteristics.
As examples for soft robots, a wrist exoskeleton that consisted of a base, an end-effector, two linear actuators, and two elastic spring blades, was developed by T. Higuma et al. [19]. The linear actuators are fixed on the base part, while the two spring blades are attached to both the end-effector and base by passive rotational joints. L. Sutton et al. [20] had developed a 1-DOF rehabilitation robot for wrist joint (flexion/extension). This wearable robot used 16 Nylon strings as the actuators and used a PID control algorithm for controlling the joint movement. Another robot was designed to rehabilitate the wrist joint by Y. Xu [21]. This robot used four new soft pneumatic actuators with molded elastomeric chambers and shell reinforcement plates.
A power augmentation wrist joint rehabilitation exoskeleton robot was developed by H. Al-Fahaam et al. [22]. This robot was focused on the wrist joint with its 2-DOFs and used five pneumatic actuators. This robot was equipped with bending PAMs to provide an actuation for the glove [23]. A soft, bendable pneumatic exoskeleton robot with two 2D fold-based was developed and 3D printed by B. W. Ang et al. [24].
However, soft-based exoskeletons presented in the literature have some limitations. Some of them induced 1-DOF for the wrist motion, such as L. Sutton et al. [20]. Others induce the possibility of coupled motion within the 2-DOFs of the wrist joint in addition to high actuators forces like T. Higuma et al. [19], H. Al-Fahaam et al. [22]. Work relevant to introducing novel actuators focuses on evaluating the static and dynamic behavior of the actuator rather than evaluating the robot performance as a whole, like H. Al-Fahaam et al. [22] and [23] and Y. Xu [21]. Since soft actuators exhibit highly nonlinear characteristics, precise modeling of developed robots based on them is crucial for both evaluating their performance and hence adjusting the suitable rehabilitation therapy that meets the patient’s needs, which of a lack for this literature. Table 1 summarizes a comparison between the characteristics of previous soft-based exoskeletons.
A simple and promising design for a wrist rehabilitation robot was proposed by G. Andrikopoulos et al. [25] that uses four pneumatic artificial muscles (PAMs) within the 2-DOFs of the wrist, Fig. 1.
This robot has many advantages. First, it simplified the complexity of the wrist movement scenarios (flexion/extension and ulnar/radial Deviation), Fig. 1, by taking advantage of a collaborative and antagonistic PAM movement strategy, where PAMs are working collaboratively in two pairs and antagonistically between pairs. Second, it avoided the use of skeletal structure for robot development which is adaptable for different users. However, the study focused on preliminary experimental evaluation of robot performance based on a nonlinear control strategy for wrist motion. The control strategy is based on increasing the pressure in a pair of muscles by value and reducing the pressure in the other pair by the same value.
In this work, it is interested in extending the work of G. Andrikopoulos et al. [25] by proposing several modes of operation of the robot according to the rehabilitation therapy based on patient’s needs. This is done by exploring the effect of the sequence of changing the pressure into the muscles on the wrist motion and the associated torques applied on the wrist within its 2-DOFs, which depend on the dynamic characteristics of the robot. Thus, exploration of robot dynamic characteristics is a prior step towards the design and implementation of a control scheme suitable for rehabilitation therapy.
Firstly, the mechanical design of the robot is developed in SOLIDWORKS software associated with the application of stress and strain analysis, followed by the design of the pneumatic system for the operation of the actuators. After that, the dynamic model of the robot is deduced and built in MATLAB/SIMULINK software, and the parameters of the robot’s different components, like the pneumatic muscles, have been experimentally identified. Two different operation modes are produced to evaluate the dynamic behavior of the robot. Finally, a design of a PD controller with feed forward acceleration is introduced.
The paper structure is as follows: the mechanical design and the actuating pneumatic circuit are introduced in Sect. 2. Mathematical modeling of the robot and the associated components are deduced and built up on MATLAB/SIMULINK software in Sect. 3. Section 4 deals with the experimental set-up and the developed algorithm for the PAMs identification. Modes of operation and the controller design are given in Sects. 5 and 6 respectively. Results and discussion are introduced in Sect. 7. Conclusions and future work are presented in Sect. 8.
In this section, the development of the wrist joint rehabilitation exoskeleton robot is discussed. Firstly, design considerations for an upper extremity motion of a robotic exoskeleton system are introduced. Secondly, details of the proposed design are introduced, then stress and strain analysis for specified parts of the robot are discussed. The pneumatic circuit used for actuating the pneumatic actuators is illustrated after that.
The design of the proposed exoskeleton robot is done according to the design consideration [26, 27]. They could be defined as:
Design Constraints: The user parameters such as human forearm length, hand length, limb weight, and range of movements of joints are important in designing the exoskeleton robot [28, 29]. Besides, the exoskeleton robot must be able to simulate the range of motion of the wrist, Table 2.
Weight and Mass Moment of Inertia: The exoskeleton robot should be light in weight to minimize the effects of inertia and gravity load. Meanwhile, the robot should have a rigid structure to support the human arm and provide the required movements at the same time.
Wrist joint torques: the applied torque on the wrist must not exceed 4 Nm for the passive wrist patient who suffers from complete wrist dysfunctionality [8].
The design of the proposed exoskeleton robot is done using SOLIDWORKS software. The robot structure is being developed to produce the 2-DOF movements of the wrist, while incorporating the minimum number of hard materials to guarantee the lightweight requirement.
Figure 2 shows the proposed exoskeleton robot with actuators which are of the flexible pneumatic type. It provides the two motions of the human wrist, the wrist flexion/extension, and the wrist radial/ulnar deviation. The robot consists of two main parts, the front and the back. The front part is movable and attached to the hand, while the back is fixed to the forearm.
The used actuators are pneumatic artificial muscles (PAMs) and arranged as shown in Fig. 2. They are made of pure rubber latex material covered with a double helical braided shell and ended by appropriate metal fittings. When compressed air is applied to the interior of the rubber tube, it contracts in length and radially expands [30]. In the proposed design, four actuators are used, two of them are placed on the robot’s upper part, while the remaining two actuators are placed downward. Resulting in a symmetrical arrangement of the robot.
According to the required motion of the hand, some of the pneumatic muscles are actuated. For example, the upper two muscles are actuated to give the hand extension, while the outside two muscles relative to the human body are being actuated to give the hand ulnar deviation, see Fig. 3. The four muscles are attached to both the front and the back parts via rotational joints.
The front part consists of two symmetrical divisions connected to each other with two flexible stretches to adopt varied sizes of the hands, as detailed in Fig. 4. Each part is 50x150x10 mm in width, length, and thickness as average dimensions, and has two circular slots of 22 mm diameter for fixing the two rotational joints supporting the actuators.
Figure 4-a illustrates the joint assembly, which consists of a U-shaped base connected to the front part via a roller bearing (22 mm and 10 mm for inner and outer diameter, respectively). The two sides of the U-shaped base have circular slots of 10 mm to place a traverse pin in between via roller bearings (10 mm and 5 mm for inner and outer diameter, respectively). The traverse pin has a circular ring in the middle to hold one terminal of the actuator by a nut, while the other terminal is held by a similar arrangement fixed to the back part of the robot. Such arrangement of the joint gives the front assembly great flexibility with the hand movement in the two directions of motion.
The back part has the same construction as the front part. It also consists of two symmetrical parts connected with six flexible stretches to guarantee stiction to the forearm, Fig. 4-b. Each part is \(130\times150\times10\text{ mm}\) in width, length, and thickness as average dimensions. The two back joints have the same construction as those of the front part with larger dimensions. The middle joints are just used as a guide for the actuators.
The augmented Polylactic Acid (PLA+) is selected as the material for all parts. It has a yield stress of 70 MPa, which results in a low mass and inertia in addition to high strength of the robot. The overall weight of the robot is 0.989 kg.
The U-shaped rotational joint is the most important part that must be examined against different actuation forces. Figure 5 illustrates the stress and strain results of the U-shaped joint. The base of the joint is fixed and a 300 N pulling force is applied to the circular ring joint (part 4 in Fig. 4). The maximum stress is found 66.5 MPa, which is less than the permissible stress.
Figure 6 shows the schematic diagram of the pneumatic system used for robot actuation. It consists of an air compressor, a pressure regulator, four electrically actuated 5/3 directional control valves (DCVs), four one-way flow control valves, and the four PAMs.
The compressed air from the compressor is regulated with a pressure regulator. The mass flow rate of the regulated air is adjusted by the one-way flow control valves. Air is directed to the PAMs by 5/3 DCVs, which control the direction of the air input/output to the muscles.
The PAM has one port for either inlet or outlet of the air. It is fixed at both ends with the front and back parts of the robot. Two PAMs are always working together, either charging/discharging, so the actuation circuit is represented by only two PAMs, with their corresponding valves of each muscle, as depicted in Fig. 5. Since upon the actuation of a pair of muscles while releasing the other pair, it is possible to obtain the motion of the hand in a specified direction. The PAMs arrangement in that way resembles the human biceps and triceps muscles.
Figure 7 describes the equivalent block diagram of the robot. Robot modeling includes modeling of the pneumatic system, the PAMs, and the mechanical construction. The robot dynamic model is then built using MATLAB/SIMULINK.
The utilized pneumatic components, which are of interest for the description of their dynamical behavior, are both the flow control and directional control valves. The directional control valve is just used for controlling the air flow path throughout the circuit and is considered to have a small time constant, from which its dynamics could be neglected, [31, 32]. Thus, this part focuses on describing the dynamic model of the air flow control valve as a relation of the air mass flow rate across the valve in terms of the valve’s upstream and downstream pressures.
Since the air flow control valve is considered as an orifice with an adjustable area, the relations of the orifice are valid for it [32, 33]. Mass flow rate through the one-way flow control valve is estimated based on Bernoulli equation, continuity equation, and critical conditions of the flow nozzle [32, 33]. Subscript 1 in the equation refers to the upstream conditions at the valve entrance, and 2 to the downstream conditions at the valve exit. The air mass flow rate through the one-way flow control valve in terms of the pressure ratio for both subsonic and sonic conditions is formulated as [34]:
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$$\begin{aligned} & \mathrm{For}\ \mathrm{Sonic} \frac{P_{2}}{P_{1}} \leq 0.528 \dot{m} =0. C A_{2} \frac{P_{1}}{\sqrt{T_{1}}} \end{aligned}$$ (1) $$\begin{aligned} & \mathrm{For}\ \mathrm{Subsonic} \frac{P_{2}}{P_{1}} >0.528 \dot{m} =0. C A_{2} \frac{P_{1}}{\sqrt{T_{1}}} \left ( \frac{P_{2}}{P_{1}} \right )^{{1} / {\gamma}} \sqrt{1- \left ( \frac{P_{2}}{P_{1}} \right )^{{\left ( \gamma -1 \right )} / {\gamma}}}, \end{aligned}$$ (2)where \(P\), \(C\), \(A\), \(\gamma \), \(T\), and \(\dot{m}\) are the pressure, orifice constant, orifice area, the air specific heat ratio, air temperature, and the mass flow rate through the valve, respectively.
The block diagram given in Fig. 6 shows that the PAMs model is divided into three relations. One is the PAMs static model, the other is the PAMs dynamic model, and the third is the PAMs volume model.
The relation of the tensile force upon the pressure application and the corresponding contraction represents the static model of the PAMs. There are many parameters that decide the static model of the PAMs, like the muscle diameter, length, and material properties, see Fig. 8. The PAMs static models are function of these variables [30, 35–38].
A previous work, focusing on introducing the relation describing the applied pressure and the corresponding contraction of the muscle to the axial force, is done in [39]. The original methods of modeling were based on the geometry of the muscle, like these in [30]. Sarosi et al. [40, 41], introduced a new approximation relation with six constant coefficients for the force \(F\) generated by fluidic muscles in terms of only the muscle pressure \(P\) and contraction ratio \(\varepsilon \) given as:
$$ F \left ( \varepsilon , P \right ) = \left ( a.P+b \right ). exp^{c.\varepsilon} +d.P.\varepsilon +e.P+f, $$ (3)where \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) are constant, their values will be obtained in the next section based on the experimental measurements. The contraction ratio \(\varepsilon \) is defined as the ratio between the muscle initial \(L_{0}\) and final \(L\) lengths to the initial length, defined as:
$$ \varepsilon = \frac{L_{0} -L}{L_{0}}. $$ (4)The pressure changes within the muscle with the air mass flow rate, either by charging or discharging, represents the PAMs dynamic model. In previous work [42–45], the authors derived this equation using the assumption that the charging process is adiabatic while the discharging process is an isothermal one. Based on the assumptions of the air gas is perfect one, the pressure and temperature within the muscle are homogeneous, and kinetic and potential energies are negligible, the PAMs dynamic model is represented as follows:
$$\begin{aligned} & \mathrm{For}\ \mathrm{charging} \dot{P} =\gamma \frac{RT}{V} \left ( \dot{m}_{in} - \dot{m}_{out} \right ) -\gamma \frac{P}{V} \dot{V}. \end{aligned}$$ (5) $$\begin{aligned} & \mathrm{For}\ \mathrm{discharging} \dot{P} = \frac{RT}{V} \left ( \dot{m}_{in} - \dot{m}_{out} \right ) - \frac{P}{V} \dot{V}, \end{aligned}$$ (6)where \(P\), \(\dot{P}\), \(R\), \(T\), \(\gamma \), \(V\), and \(\dot{m}\) are the absolute pressure, pressure variance, air gas constant, air temperature, air specific heat ratio, and volume of the muscle, respectively.
Finally, the PAM volume model can be obtained by considering the muscle as a cylinder with zero wall thickness, Fig. 8, its volume could be given as:
$$ V= \frac{1}{4} \pi D^{2} L= \frac{B^{3} \cos \alpha \sin ^{2} \alpha}{4 n^{2} \pi}, $$ (7)where, \(D\), \(L\), \(B\), \(n\), and \(\alpha \) are muscle inner diameter, muscle length, thread length, number of turns, and the angle of the threads with the longitudinal axis, respectively.
The PAMs can generate force only in contraction, so the muscles should be grouped into pairs for bidirectional motions like human muscles (biceps and triceps). The rotational motion of a plant is determined by the actuating forces exerted by two PAMs, as shown in Fig. 6. If the actuating forces for each muscle are denoted as \(F_{1}\) and \(F_{2}\), the equation of motion for the antagonistic pair is (based on Newton’s equation \(\sum F =I\alpha \)).
$$ I \ddot{\theta} + B_{v} \dot{\theta} = \left ( F_{1} - F_{2} \right ) r, $$ (8)where, \(I\) and \(B_{v}\) are the mass moment of inertia of the rotating part (i.e., the front part and the hand in our case) and the viscous damping coefficient for the used bearing, respectively. \(\theta \) is the rotational displacement of the antagonistic structure and \(r\) is the equivalent radius of the joint. Since, the actuation forces \(F_{1}\) and \(F_{2}\) are determined by the pressures applied to the muscles, the joint displacement can be adjusted by controlling input pressures.
Figures 9 and 10 represent the block diagram of the robot and the simulation model on SIMULINK, respectively. Index 1 in the model represents the first muscle (PAM-1) and index 2 refers to the second muscle (PAM-2). Table 3 gives the parameters used in the simulation.
The results of the PAMs identification, dynamic evaluation of the two actuation modes, and the control results are presented in this section.
The identification results based on Eqn. (3) are displayed on Fig. 16. The parameters defined in Eqn. (3) are optimized based on the experimental data of the measured forces and the corresponding contractions at different applied pressures based on the least square error approach using the user-defined function (lsqnonlin) in MATLAB. The optimization options are set to 10−12 squared error and 10 times multi-start to ensure the optimized values of the parameters.
As seen in Fig. 16, there is an acceptable matching between experimental and fitted data in general, except at some small load values at various applied pressures. The optimized values of the parameters \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) in Eqn. (3), are given in Table 4. The established cross-correlation index between the experimental and calculated forces for this strategy is mentioned in [39] and is given as (R = 0. and R\(^{2} = 0.\)).
Figure 17 represents the pressure and angle results for the extension/flexion motion of the wrist joint upon the operation of mode-1. The PAMs are initially inflated to 6 bar absolute pressure (\(P_{\mathrm{init}} =6\ \mathrm{bar}\)). The upper PAM is discharged to the atmospheric pressure (1 bar-absolute), blocked for a while, and re-charged again, while the lower muscle is kept blocked throughout the discharging and charging processes. This is done at four different actuation timings (1.5 sec, 1 sec, 0.5 sec, and 0.3 sec). The angle of the wrist joint increases as the muscles are discharged reaching its maximum near the blocking timing.
Figure 17-a shows the pressure variance inside the upper PAM during discharging, blocking, and charging again to the same pressure (6 bar). The results depicted that the muscle needs 1.5 sec to reach the full discharge (reach 1 bar-absolute) at the addressed capacitance coefficient of the flow control valve. When the actuation timing decreases less than 1.5 sec, the muscle pressure does not reach the minimum pressure. For 1 sec, 0.5 sec, and 0.3 sec actuation timings, the muscle pressure reaches 1.6 bar, 3.2 bar, and 4.2 bar, respectively during the discharging, and reaches 5.95 bar, 5.45 bar, and 5.38 bar, respectively during the charging process.
Figure 17-b shows the pressure variance inside the downward PAMs that are blocked from the beginning till the end of simulation. This figure illustrates that the muscle pressure decreases as the upper muscles are discharged, being constant throughout the blocking time and re-increases again during the charging time. For 1.5 sec, 1 sec, 0.5 sec, and 0.3 sec actuation timings, the lower muscles pressure reaches 4.26 bar, 4.4 bar, 4.92 bar, and 5.28 bar, respectively during the discharging of the upper muscles, and reaches 6 bar, 5.98 bar, 5.78 bar, and 5.75 bar, respectively during the charging process.
Wrist angle results are illustrated in Fig. 17-c. As the upper PAMs discharge, the wrist angle increases in the negative direction, i.e., towards the blocked muscle, Fig. 5. The wrist joint angle reaches its maximum value at the blocking time. This value decreases at lower actuation timing. The wrist angle reaches −9.45°, −7.98°, −4.9°, and −3° at 1.5 sec, 1 sec, 0.5 sec, and 0.3 sec actuation timing, respectively during the upper muscles discharging.
Figure 18 shows the pressure and angle results of the second DOF of the wrist joint, the radial/ulnar motion. The pressure results are the same as the previous motion, as shown in Figs. 18-a and b, but the angle results show little difference in the angle values with the same trend as the other DOF. The angle of the wrist joint for this motion (radial/ulnar) becomes −4.5°, −4.1°, −2.45°, and −1.5° with actuation timing of 1.5 sec, 1 sec, 0.5 sec, and 0.3 sec, respectively.
Figure 19 describes the difference between the muscle pressures at the beginning and end of either the charging or the discharging processes for both muscles at different actuation timings. It is noted that the pressure difference during the discharging is larger than that during the charging for the same actuating time. This means that the muscle needs more time for charging than discharging for the same pressure difference.
Figure 20 shows the maximum difference force at the different actuation timing for the writ motion within its 2-DOFs. The maximum difference force is about 3.62 N, resulting in a torque of about 0.22 Nm for the flexion/extension motion, while the maximum difference force is 2.8 N, resulting in a torque of about 0.08 Nm for the other motion. The torque values are calculated based on the radii of rotation given in Table 3.
Figure 21 shows the maximum difference force at different initial absolute pressures (from 2 bar to 8 bar) for the same actuation timing of 1.5 sec for only the ulnar/radial motion. The results of the muscles pressure and the wrist angle are not shown since they have same trend as in Fig. 18.
As it appears from Fig. 21, increasing the muscles initial pressure gradually increases the maximum difference force and subsequently the wrist joint torque and range of motion. Clearly, this mode exhibits great flexibility to move the wrist joint to a specified angle while keeping the wrist joint torque at some specified value. This could be achieved by controlling both the muscles initial pressure and the actuation timing.
Effect of the wrist joint characteristics
The characteristics of the wrist joint are included into the model to validate its effectiveness. This is implemented during the flexion/extension motion at the four actuation timings 1.5, 1, 0.5, and 0.3 sec, respectively. The damping coefficient is taken as 0.06 Nms/rad [46, 47]. However, the wrist joint stiffness is not constant, it varies according to the grasping force [46, 48, 49]. As an average value it has been considered as 3 Nm/rad.
Including the joint stiffness and damping characteristics are done by re-formulating Eqn. (8) to include both the mechanical construction of the robot as well as the human wrist joint, to be on the form:
$$ I \ddot{\theta} + \left ( B_{v} + B_{j} \right ) \dot{\theta} + k_{j} \theta = \left ( F_{1} - F_{2} \right ) r, $$ (11)where \(k_{j}\) and \(B_{j}\) represent the wrist joint stiffness and damping throughout the flexion/extension motion, respectively. Figure 22 shows the wrist joint angle results upon considering Eqn. (11) instead of Eqn. (8). As it appears, the trend of change is almost the same compared with the wrist angle results based on Eqn. (8), Fig. 17-c. The steady-state value that the wrist joint reaches either upon chagrining or discharging is very near to that of Fig. 17-c. Figure 23 shows the steady-state values based on Eqns. (8) and (11) at both chagrining and discharging. They are also given, as well as the absolute error, in Table 5. The max. absolute error is 0.146°, that exhibits the possibility of the dependency of Eqn. (8) for the study of the dynamical behavior of the human- robot interaction.
The purpose of this mode is to investigate the difference force of the muscles (and hence the wrist joint torques) upon reaching one muscle to the maximum charging pressure while the other one reaches its minimum at different frequencies.
In this mode, one PAM is charging while the other PAM is discharging at the same time. This is done at different frequencies (0.5 Hz, 5 Hz and 15 Hz). This simulation is done with 6 bar initial pressure. The results are shown only for the flexion/extension motion.
Figures 24, 25 and 26 show the pressure, wrist joint angle, and difference force results for the second mode actuation (0.5 Hz, 5 Hz, and 15 Hz) respectively.
For the frequency rate of 0.5 Hz, the angle reaches 12.3° in both directions. As the frequency increases, the maximum pressure inside the PAMs decreases and reaches 5 bar and 4.2 bar for 5 Hz and 15 Hz actuation frequency, respectively. Also, the wrist joint angle becomes 6.5° for 5 Hz and 3.5° for 15 Hz, as shown in Figs. 25-b and 26-b.
The differential force is kept around 52 N as a max. value for all frequencies. This means the wrist joint torque is about 3 Nm for the flexion/extension motion that is within the permissible range. This is obtained during the charging and discharging processes. Once the muscle pressure reaches its limits within the two muscles, the difference force is kept at its minimum limit. It is about 2.7 N for 0.5 Hz actuation frequency.
A remarkable note is, although upon increasing the frequency of actuation the wrist joint range of motion decreases because the muscles don’t have enough time to reach their upper and lower pressure limits, the maximum wrist joint torque is kept almost constant at the beginning of motion. This means that this mode is suitable for the therapy of passive patients who have complete wrist joint dysfunctionality where more joint torques are required. Furthermore, this mode exhibits a greater range of motion compared to mode-1 for the same muscle pressure limits and mass flow rates.
The PD controller with feedforward acceleration has been designed for tracking the required trajectory of 10° wrist angles over 2 sec. The operation of the controller is based on mode-1 of operation, i.e., the pressure varies in one muscle while the other muscle pressure is kept constant at the minimum value (the atmospheric pressure). Figure 27 exhibits simulation results. The controller is evaluated at the two models based on Eqns. (8) and (11). The maximum angle error is 0.32° and 0.069° for Eqns. (8) and (11), respectively. As a result, the proposed control system is capable of tracking specified hand trajectories for different wrist joints characteristics.
As a comparison to previous related work in Table 1, the proposed robot covers a wider range of applied torques at the wrist joint, ranging from 0:3 Nm. The covered range is within the permissible values of the required torques for wrist patients (0:4 Nm, [8]). However, it is limited in terms of the lower range of motion (max. obtained angle in ext./flex. 12.3°). A larger range of motion could be covered once the actuators’ difference force as well as the actuating timing are increased.
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