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A 3-DoF Robotic Platform for the Rehabilitation and Assessment of Reaction Time and Balance Skills of MS Patients

The central nervous system (CNS) exploits anticipatory (APAs) and compensatory (CPAs) postural adjustments to maintain the balance.The postural adjustments comprising stability of the center of mass (CoM) and the pressure distribution of the body influence each other if there is a lack of performance in either of them.Any predictable or sudden perturbation may pave the way for the divergence of CoM from equilibrium and in homogeneous pressure distribution of the body.Such a situation is often observed in daily livings of Multiple Sclerosis (MS) patients owing to their poor APAs and CPAs, and induces their falls.The way of minimizing risk of falls in neurological patients is utilizing perturbation-based rehabilitation, as it is efficient in the recovery of the balance disorder.In the light of the findings, we present the design, implementation, and experimental evaluation of a novel 3 DoF parallel manipulator to treat the balance disorder of MS.The robotic platform allows angular motion of the ankle based on its anthropomorphic freedom.The end-effector endowed with upper and lower platforms is designed to evaluate both the pressure distribution of each foot and the CoM of the body, respectively.Data gathered from the platforms are utilized to both evaluate performance of the patients and used in high-level control of the robotic platform to regulate the difficulty level of tasks.In this study, kinematic and dynamic analyses of the robot are derived and validated in the simulation environment. Low-level control of the prototype is also successfully implemented through PID controller.The capacity of each platform is evaluated with a set of experiments considering assessment of pressure distribution and CoM of the foot-like-objects on the end-effector. Experimental results indicate that such a system well-address the need for balance skill training and assessment through the APAs and CPAs.


Introduction
Multiple Sclerosis (MS) is one of the most common chronic neurological diseases in the young adult age group, and influences around 2.5 million people in the world. As it affects brain, CNS, and spinal cord of a patient, MS is the most common disease-causing disability due to neurological reasons. In particular, more than seventy-five percent of MS patients experience symptoms of balance disorder [1], sixty percent of them encounter at least one fall in 3 months, and more than eighty percent report activity restrictions in their daily lives. Such situations promote a sedentary lifestyle, reduce community and social participation, and lower living standards of the patients [2,3].
Balance disorder, one of the prominent symptoms due to lesions in the sections of the brain responsible for movement and balance, is common in MS patients [4][5][6]. In particular, reaching a stable gate pattern takes longer time for the patients, and they inevitably do experience recurrent fall events. In healthy people, in order to maintain the balance of the whole body, central control of posture utilizes the anticipatory (feed-forward) and compensatory (feedback) postural adjustments (APAs and CPAs). APAs engage and activate the trunk and lower extremity postural muscles before an impending condition/perturbation occurs. It reduces the risk of deterioration in balance by regulating the position of the CoM. CPAs are triggered by the sensory control signal and allow the CoM to be repositioned once it has been disrupted. CPAs are activated in both predictable and unpredictable conditions further; when APAs are not able to ensure balance condition, the body is dependent only on CPAs. These postural adjustments work as coupled, and if one has a deficiency, the other is also affected [3,7]. The MS patients suffer from insufficient balance control due to their poor APAs and CPAs. Hence, findings in the literature state that perturbation-based rehabilitation programs can improve the balance and decrease the falling rate of MS patients. For instance, the patient may expose to a postural perturbation without any prior information about it, like the robot-based rehabilitation, or the source of the perturbation may become a physioterapist and he/she can disturb the patients by throwing them a ball [8,9].
The other common symptom to diagnose and follow up during the therapy and treatment phase of the MS patients is their reaction time against the intentionally provided external stimulation [10,11]. Basically, the reaction time is the period that elapses from the start time of the stimulus and the time interval from which the response begins. The reaction time is often used to assess the sensory-based actuator disorders and indicates the balance disorder's severity. In particular, MS patients take 1.5 times longer time to stabilize their posture and reach safer state in a longer period than their healthy pair, which leads to inevitable falling [6]. As the longer reaction time is an indication of the loss of balance control, MS disease severity is evaluated and treated as a means of patients' efforts to perturbations. Along these lines, some research groups [12] have focused on the APA capability of the patients. In this particular, they conducted human-subject experiments on fallers and non-fallers MS patients and observed electromyography(EMG) activity of their leg muscles and distribution of the center of pressure. The experimental evaluation reveals that the MS patients with a fall history had weaker electrical activity and a correspondingly long reaction time to regain their balance with postural adjustment.
The efficacy of the rehabilitation on MS patients was also proved with a task-based evaluation. In this study [8], distortion of their CoM was measured by means of the platform where the patients stood on and the EMG activity levels were evaluated while the patients were catching thrown ball randomly. After regular training, patients were able to keep their CoM in the close vicinity of the equilibrium point and regain their balance ability. Findings from this study provide a background for the development of perturbation-based rehabilitation programs to improve balance and prevent falling in patients.
MS patients in the mild-to-moderate disability range are recommended to receive regular rehabilitation therapy four times a week in order to adapt to daily life more easily [13]. Despite such an essential need, findings in the literature show that currently, only thirty percent of MS patients benefit from rehabilitation services, and those who could not receive therapy have to use a cane within 20 years and a wheelchair within 30 years from the onset of symptoms [14]. Due to the reason, demands for therapeutic rehabilitation robots have increased and the research in this field has grown rapidly as robot-assisted rehabilitation can deliver efficient training and provide objective follow-up assessment and evaluation relative to the conventional rehabilitation techniques [15]. As for MS syndrome, the research on assistive devices has mainly focused on unilateral or bilateral upper limb dysfunctions since around fifty percent of people with MS suffer from impaired hand functions [16,17]. Some research groups [18,19] developed a robotic system, called Haptic Master accompanied with virtual learning environment (I-TRAVLE) for the MS rehabilitation of upper limb dysfunction. Patients forced their impaired arm by means of the assistive system and performed the tasks defined in the virtual reality (VR) environment. Findings showed that actuator control ability of the patients were improved and allowed to lift up their arm to higher level after a regular training program. Upper limb exoskeletons were also proposed to assist MS patients with paresis [20]. In the study, 5 Degree of freedom (DoF) Armeo Spring (Hocoma AG, Zurich, CH) was employed to strengthen the muscle groups of the patients. This pilot study provided evidence that 8-week training with the Armeo Spring and improved the patients' upper limb muscle strengths and functional ability.
In the literature, some research groups have also focused on the lower limb rehabilitation of people with MS. For instance, ankle rehabilitation is considered as an effective way to improve walking performance, muscle strength [15] as well as range of motion of the ankle of MS patients [21]. To address the problem, researchers mainly have utilized two fundamental concepts developed in robotic rehabilitation. Lower limb exoskeletons have been employed for years to treat the neurological disorders. In the study [22], the exoskeleton together with the assist-as-needed concept was used to explore the efficacy of the MAK exoskeleton for gait rehabilitation of MS patients. Results showed that MAK exoskeleton improved the gait performance of the patients to a some extend; however loss of stability was observed during the training. On the other side of the ankle rehabilitation, the robotic platforms have served to improve walking patterns of MS patients. In the study [23], 1 DoF robotic platform (RePAiR) allows to realize dorsiflexion and plantar flexion movements of the post-stroke patients. The comparison between the patients and the control group revealed that the device availed to increase the muscle strength, improved actuator control, sensory-actuator coordination of patients, and accordingly walking patterns. A 2 DoF robotic platform [24] was also proposed to train ankle in plantar/dorsiflexion and inversion/eversion directions. However, such platforms are specifically designed for the ankle rehabilitation and do not treat the balance dysfunction in MS.
Posturography, the expression of postural balance or sway, is evaluated basically by means of platforms with integrated load cells that can provide pressure/force measurement. Such evaluation can be implemented in either static or dynamic means. For the static case, it is expected from patients to stand on the fixed platform and fulfill the required tasks displayed. While some are doing tasks by means of physical interactions, like throwing a ball while standing on the platform, the others are performing tasks introduced in VR environment in order to evaluate their center of mass and pressure distribution [25]. Some commercially available platforms [26][27][28][29] exist to assess the performance of the patients under static conditions. Such devices have been put into the services of patients together with the VR environment to give visual feedback and commands, and accordingly prolong the training period [4]. Although it has been stated that training with static balance platforms is effective in the regulation of pressure distribution in patients in a longer time [30], the findings indicate that rehabilitation of balance disorder in dynamic conditions contributes significantly to the recovery of actuator functions [26].
Dynamic platforms have been designed to treat balance disorders, like the static ones; however such systems require instantaneous dynamic action of the patients by forcing them to regulate their balance even under perturbations. Various studies have been performed to regulate the balance stability of people. For instance, balance boards [31] have been developed either electronically or by wood plate placed on a roller to train the people. The ones including load cells are able to measure the pressure of the foot and center of gravity of the patient [32][33][34][35]. Another solution to train the patient has been proposed by GRAIL [36], which is a dynamic treadmill working in harmony with the VR environment. Some commercially available robotic platforms [37,38] have been produced to address the post-traumatic, orthopedic or neurological problems by training hip, ankle, and shoulder joints. Depending on the instruction given by the therapist to the robot, the robotic system can intentionally expose patients to an unbalanced state and expect them to strive against the instability. Such systems are able to measure the pressure distribution of each foot and evaluate balance level by means of the center of mass of the body. Human-subject experiments provided evidence that dynamic force platforms help to regain dexterity of the limbs and postural stability compared to the conventional treatment group [39]. Despite the benefits of the platforms, there is no study conducted in the literature to investigate the ability to control APAs and CPAs, accordingly the reaction time of the patients, and regulate the control strategy of the platform based on such information. Moreover, the aforementioned robotic platforms were designed as a series of links connected by actuator-actuated joints. Although it is easier to model and do the dynamic analyses of the serial manipulators, preferring parallel manipulator in such an application provides advantages in terms of achieving high load-carrying capability, better dynamic performance, and precise positioning. Considering the efficacy of the parallel manipulator, to the best of our knowledge, there is no such study developed as a parallel manipulator to carry out balance training in the literature.
In this study, we present the design, simulation, implementation and experimental evaluation of a 3 DoF parallel manipulator to rehabilitate the balance dysfunctions of the MS patients. The proposed platform is basically composed of three main parts: 3 DoF parallel manipulator, upper and lower platforms. The 3 DoF parallel manipulator is able to provide both rotational (roll, pitch angles) and linear motion (z direction) in space and also intentionally destabilize the human body via perturbations in order to force them to regulate their balance. The reactions of the patients to maintain their balance while the manipulator is moving are evaluated through the upper and lower platforms. The upper platform is responsible for evaluating the pressure distribution of the feet to follow up their balance capacity and measuring the reaction time of human against the predictable conditions and perturbations. The lower platform designed to analyze the CoM of the patients in real-time and points out the severity level of the MS patients. Moreover, the information acquired from the upper and lower platforms is employed as biofeedback to adjust the assistance level required by the assist-as-needed paradigm used in the control architecture of the planar manipulator.
The rest of the manuscript is organized as follows: Section II introduces the design objectives of the robotic platform, presents kinematic and dynamic analyses of the 3DoF parallel manipulator and its simulation, and also details the design of the upper and lower platform of the robot, while Section III evaluates the manipulator's performance and capability of the upper and lower platforms to analyze the center of mass and pressure distribution of the objects in both static and dynamic conditions. Finally, Section IV concludes the paper and presents the discussion and future work.

Materials and Methods
This section presents the design objectives of the 3 DoF parallel manipulator to be used for the MS patients, kinematic and dynamic analyses of the system and evaluation of the overall system by means of simulation results.

Design Objectives of 3 DoF Robotic Platform
The performance requirements of the robotic platform for the rehabilitation of MS patients are categorized based on the presented terminology in [40], of imperative, optimal, primary, secondary, and tertiary requirements.
Anthropomorphic compliance is an imperative design requirement for the 3 DoF robotic platform. Enabling patients to exercise within the limit of ankle freedom based on the disorder level is both essential for patients' safety and raising the efficiency of the activity. Besides, doctors/physiotherapists are allowed to adjust the angular constraints of the platform depending on the upper and lower limits of the patients' ability. Accordingly, considering the healthy human, the capacity of the ankle is identified in three directions with the following ranges: [41].
Designing the platform capable of measuring and evaluating the reaction time of the patients during the activities is the optimal performance requirement. In particular, considering the balance problem of the MS patients, the variation of the center of mass (CoM) of their body in the cartesian coordinates is observed more dramatically under both static and dynamic conditions relative to healthy people. Moreover, the unbalanced body gives rise to the unbalanced distribution of the load delivered by the feet. Hence, the proposed design should allow doctors/physiotherapists to evaluate the balance condition of the patients as well as their cognitive and physical reactions to the physical perturbations applied by the robotic platform itself to compensate for the disturbed balance. Accordingly, the proposed system provides quantitative evaluation about the robustness of the patients against the physical disturbances while this robotic platform accompanied by the VR environment helps the patients recover their neuromuscular system.
The primary design requirement of the platform is identifying the CoM of the body and pressure distribution of each foot based on the anthropomorphic size of each person. During rehabilitation, instantaneously measured CoM of the patients gives biofeedback to the robotic platform about the performance and severity level of disorders, and thus it allows the regulation of the assistance level in control architecture to assist the patients as needed. Moreover, the pressure distribution of the feet indicates the intention of patients to balance their body under disturbances, and accordingly provides information about the muscular activity during the follow-up. The end-effector of the parallel manipulator is designed as ergonomic and personalized two-fold layers that embody multiple load cells. In particular, the upper layer, named the upper platform, is endowed with eight load cells which are placed on the sliders to adjust their location depending on any foot size of the patients. Besides, the second layer, named the lower platform, includes three load cells positioned to form an equilateral triangle, that allows to detect the CoM of a patient in x-y coordinate plane.
The secondary requirement for the overall system design is to guarantee the safety of the patients standing on the platform under dynamic conditions. The 3 DoF parallel manipulator is designed to be able to carry the average human weight, around 687N, as well as the weight of the end-effector (lower and upper platforms), around 120N. That being said, each linear actuator has 900N force capacity to resist against more than the maximum load and are positioned under the end-effector with equal distance as seen in Figure . In order to ensure the safety of the system and its performance, the kinematic and dynamic analyses of the platform have been implemented both in simulation environment and in real environment with the proof of concept design. Figure 1 .
The tertiary requirement of the system's design criteria is increasing the efficiency of the training to get higher benefit from rehabilitation. To address the need of the criterion, the physical system is accompanied with the VR environment such a way that the patients are both motivated by means of the tasks defined in the VR and conducted in a correct manner to do activities properly. The VR environment as depicted in Figure a is placed in front of the patients to satisfy body-eye coordination properly. The task in the VR environment is analogously designed with the activities forced by the robotic platform. For instance, the angular deviation of the end-effector is matched with the greenish ball, a.k.a. haptic proxy, presented in Figure b. In this example, it is expected from the patient to keep the green ball in the oscillatory red cylinder which moves up and down with random frequencies by controlling the angular position of the end-effector. While the red cylinder is perturbed, the patient physically feels the same action by the robotic platform and responds to it to control the haptic proxy. During such activities, the system is able to measure the reaction time of the patient's attempt to the physical perturbations. Moreover, measured data allow us to quantitatively evaluate the patient's anticipatory postural adjustment in his/her intention phase to maintain the balance of the posture. Figure 2 .

Kinematic and Dynamic Analyses of 3 DoF Parallel Manipulator
In this section, we detail the kinematic analysis of the 3 DoF parallel manipulator constructing the vector loop equations and the system's dynamic analysis based on the Euler-Lagrange method.

Kinematic Analysis of the Manipulator
Figure presents the schematic representation of the 3 DoF parallel manipulator where the bold lines describe the base of the platform. Two DoF of the manipulator, roll(α), pitch(β) angles, are assigned for the orientations in two directions, while one DoF of the manipulator is allocated for the translation in z direction. The proposed solution for the kinematic analysis utilizes three vector loop equations to construct the geometric relations among the frames. As depicted in Figure , the end-effector and the base frame are geometrically expressed as a means of two equilateral triangles connected by frames of three linear actuators. In other words, the corners of the triangles typify the ball and socket joints transferring the force generated by the linear actuators to end-effector. The main reason of selecting ball and socket type joints in such a system is to ensure that while one linear actuator moves but the others are not affected. Here, the symbol r represents the distance between the corners of the triangles and the center of the platform, also symbol θ i defines the positioning of the actuators on the base frame and is fixed to 70 • . Symbols X and u are the centers of base platform and upper platform (end-effector) as depicted in Figure , respectively. A fixed coordinate frame (X) is established on the base platform and the coordinates of the frame are named x 1 , y 2 , and z 3 .
Coordinates A 1 , A 2 , and A 3 of the corners of the base frame are interpreted with respect to X frame by means of the position vectors in expression 1.
The coordinates of the corners of upper platform depicted in Figure , b 1 , b 2 , and b 3 with respect to u frame are described in (2).
The frame u can be expressed with respect to the X frame by using Euler Angle Representation. In Matrix (3), symbols (α), (β) and (γ) represent roll, pitch, and yaw angles, respectively. Accordingly, the coordinates, b 1 , b 2 , and b 3 stated with respect to the X frame by means of the Euler Angle transformation, T is expressed in matrix notation as: Actuators' positions on X frame identify the constraints of the kinematics as illustrated in Figure Eq (11) and Eq (12) are derived to represent the constraint equations in u frame by adding and subtracting Eq (9) and Eq (10).
Eq (11) satisfies the equality for any integer of n; however considering the physical limitation of the system requirement, the summation of α and γ cannot exceed zero as expressed in Eq (13).
The constraint Eqs (13)(14)(15) are rearranged to diminish the number of unknowns from five (x,y,roll(α), pitch(β) and yaw(γ)) to two (roll(α), and pitch(β)). The relations among the x,y,z, roll(α), pitch(β) and yaw(γ) angles expressed in Eqs (11)(12) are utilized to determine the constraint Eqs (13)(14)(15) to characterize the upper platform by X U and Y U depicted in Figure A. α + γ = nπ, where n = 0 (13) The connection between the upper and lower platforms is carried out via three linear actuators. Considering kinematic point of view, the vector based representation of the actuators' positions in the space is defined by their lengths, l i , where i = 1, 2, 3. The Pythagorean theorem allows us to establish the fundamental relation in Euclidean geometry among the three sides of a right triangle as depicted in Figure B and the formula to express the length of each actuator is presented in Eq (16).
Accordingly, the length of each actuator is expressed as a function of three independent variables of roll(α), pitch(β), and Z U .
The kinematic architecture of the parallel manipulator is constructed based on the closed-loop kinematic chains as illustrated in Figure A. In order to solve forward kinematic problem for the manipulator based on the position variation of each actuator as a function of time, three vector loop equations are derived by summing up the vectors presented in Figure B and expressed in Eq (20). Similarly, the inverse kinematics of the manipulator is formulated by rearranging the vector loop equations (21)(22)(23) to determine the instantaneous actuator position required for the end effector to reach the desired position.
July 26, 2022 9/24 The positioning of the actuators on the end-effector forming the corners of the triangular shape depicted in Figure A enables us to calculate the coordinate of the centroid of the triangle, namely the end-effector (X c , Y c , Z c ) in expression 30 by averaging the well-defined coordinates,X U , Y U , Z U .

Dynamic Analysis of the Manipulator
The 3-DoF parallel manipulator considered in this work is equipped with three linear actuators mounted between the end-effector and the base of the robot. The end-effector is able to rotate in roll (α) and pitch (β) angles and moves in z axis as represented in Figure B. The dynamic model of a 3-DoF parallel manipulator in joint space is derived utilizing a Euler-Lagrange method. Accordingly, the solutions extract the required forces/torques that maintain the system to follow the reference trajectory. The general form of dynamic equation is, where, X symbolizes the generalized coordinate vector of the manipulator, M represents mass vector, C is the centrifugal and coriolis matrix, G is the gravity vector and F is a vectorial representation of the required forces to actuate each linear motor in task space. The kinematics of the platform is structured based on the unit vector ( s i ) along actuator direction utilizing the loop equations discussed in kinematics analysis of the manipulator.
To find dynamic equation, closed loop equation and link lengths are redefined by using unit vector ( s i ) along actuator direction. s 1 , s 2 , and s 3 are unit vectors are in the same direction as axes K1, M1, and P1 shown in Figure , respectively As found in inverse kinematics, length of each actuator found as To express length of the actuators in coordinate frame, unit vectors are defined as, By differentiation Eq 35 and 36, velocities of intermediate variable can be found as, where w i andl i represent angular velocity and length rate of each actuators,respectively. Symbol w is the angular velocity of end effector. Again, by differentiation Eq (37) and Eq (38), acceleration of intermediate variable can be found as,δ Then by applying dot product between Eq (37) and s i ,angular and linear velocity of the intermediate variable can express with position of the actuators, Applying cross product procedure in Eq (37) and Eq (38) to s i , angular velocity of each actuator yield, In order to make calculations easier, dynamic analysis of the actuators and moving platforms were done separately and later combined with the Jacobian matrix.

Dynamic Analysis of Each Actuation Unit
The dynamic equation in Eq (31) stated in task space is reformed and expressed as Eq (43) to found the dynamic equation of each actuator in joint space.
where M i , C i , G i and F i represent mass matrix, centrifugal and coriolis matrix, gravity vector of the actuators and generated force by each actuator, respectively.
The assembly of the linear actuator is composed of two main parts as represented Figure C. The stationary part is responsible for power generation, the moving part is in charge of displacement of the shaft. Accordingly, the dynamic analysis of the overall action unit is derived considering the mechanical characteristics, e.g. mass, geometry and CoM, of each part. The CoM with respect to its own frame and mass of stationary part and moving part are represented by symbols c i1 , c i2 and m i1 , m i2 , respectively.
The velocities of stationary and moving part are obtained by time derivative of Eq (44) and Eq (45) and expressed in Eq (46) and Eq (47).
The skew-symmetric matrix form of a vector is represented by the symbol S(.). Accordingly, Eqs (42, 46 and 47) are reconstructed in the form of S(.) and expressed in Eqs (48, 49, 50 and 51).
Based on the Euler-Lagrange formalism, kinetic energy of the linear actuator is formed as in Eq (52).
where I ci1 and I ci2 represent the moment of inertia (MoI) of stationary and moving part, respectively.
Rearranging the Eq (52) yields the kinetic energy of the actuation unit expressed in Eq (53).
The mass matrix, M i is extracted from the kinetic energy Eq (53) and expressed in Eq (54).
Deriving the potential energy equation in Eq (55) of the actuation unit is also required to generate the dynamics analysis.
The gravity matrix is generated as in Eq (56) by deriving the potential energy equation with respect to the variable, δ i . .
Centrifugal and coriolis force matrix is derived utilizing the Euler-Lagrange formalism in Eq (57) and detailed in Eq (58).

Dynamic Analysis of End-Effector
The general dynamic equation of the end-effector of the robotic platform, the upper and lower platforms, is expressed in Eq.59.
,where M e , C e , G e and F e represent mass matrix, centrifugal and coriolis matrix, gravity vector of the end-effector, and force exerted by the end-effector, respectively. The position and orientation of the end-effector with respect to the base frame is expressed by X = [x; y; z; α; β; γ]. The time derivative of X that constitutes the velocity of the end-effector yieldsẊ = [ẋ;ẏ;ż;α;β;γ].
The dynamic analysis of the end-effector requires transforming the variables from the base frame to their own frame through R T matrix.
The angular velocity of the end-effector with respect to its own frame is expressed utilizing the Eq (60).
July 26, 2022 13/24 The angular acceleration of the end-effector with respect to its own frame is derived by means of R T and its time derivative,Ṙ T , and expressed in Eq (62).
Kinetic energy equation of the end-effector to be employed in the Euler-Lagrange formalism is formed as in 63.
,where m e and I e are mass and inertia matrix of the end-effector. Here, v e is the time derivative of Eq (30).
The mass matrix, M e is extracted from the kinetic energy Eq (63) and expressed in Potential energy equation in Eq (65) of the end-effector is needed to analyze the dynamics of the end-effector.
The gravity matrix is generated as in Eq (66) by deriving the potential energy function with respect to the vector X.
Centrifugal and coriolis force matrix is derived utilizing the Euler-Lagrange formula in Eq (67).

Dynamic Analysis of the Overall System
The intermediate variable (δ i ) configured by each linear actuator affects the position and orientation of the end-effector in task space in an independent manner. Considering the characteristics of the parallel manipulator, the configuration of each actuation unit affects the position and orientation of the end-effector's frame. Since the dynamic representation of the robotic platform, (M all , C all , andG all ), needs to be expressed in task space, dynamic equation of each linear actuator written with respect to its own frame as expressed in Item 1 has to be stated in task space through the Jacobian matrix in 64.
July 26, 2022 14/24 The dynamic contributions of linear actuators and end effectors are added together as in Eq (69, 70, and 71).
Considering the dynamic effect of each member of the system, the generated force to actuate the end-effector in the work space is calculated by the Eq (72).

Simulation of the Kinematics and Dynamics of the 3 DoF Robotic Manipulator
In this Section, the derivations expressed the kinematics and dynamics of the 3 DoF parallel manipulator are verified on the simulation environment. The overall flowchart of the simulation to represent forward and inverse kinematics and dynamics obtained in Section 2.2.1 and Section 2.2.2 is presented in Figure 4.

Design of Upper and Lower Platforms
The upper platform presented as solid model in Figure A and as a first prototype in Figure C responsible for the evaluation of the pressure distribution has two places for each foot. Each part is endowed with four load cells placed under the sole. The proper placement of the load cells is determined based on the effective pressure regions of the sole. In literature [42], the more densely felt four pressure regions of the foot sole were evaluated and the pressure values are presented in Table 1. Figure 8 . The sliders are embedded in the upper platform to adjust the load cells' coordinates to sense the pressure distribution for any foot size accurately. The sliders' strokes are determined based on the average anthropometric data of human [43][44][45] and presented in Table 2. The pressure distributions under the feet as a function of time can also be evaluated by the upper platform. Evaluation of such variation is used to calculate the reaction time of the patients during the rehabilitation. The elapsed time until the patient keeps his/her balance against the perturbation, namely CPA, is detected and considered as a performance metric for the patient's follow-ups. This metric allows the robot to regulate the level of assistance based on the assist-as-needed paradigm.
The lower platform located just under the upper platform is in charge of evaluation of the CoM of patients in both static and dynamic conditions. This measurement is utilized to indicate the balance status of the patients as well as give biofeedback to the robotic platform to adjust its assistance level for the patient. The required data to calculate the CoM of the body in x-y coordinate plane is measured by means of three load cells positioned on the corners of the hidden equilateral triangle placed on the lower platform shown in Figure B and Figure D. The CoM of the patients in x-y coordinate plane is calculated based on the Eq (73) and Eq (74). The partial weights of the body distributed over the load cells 9, 10, and 11 are named weight-9, weight-10, and weight-11, respectively, also symbol h represents the distance between each corner and the centroid of the equilateral triangle as depicted in Figure B.

Stress Analyses of the Robotic Platform
The stress analyses of the robotic platform are examined as a means of finite element analysis under the loading of average human weight (687N) and weight of the platform (120N). In Figure , the response of each main part, of upper and lower platforms, linear actuators, against the static nodal stress is demonstrated. The results show that the compression force leads to less than 1mm deformation, that is quite negligible and justifies the rigid and durable characteristics of the design. Accordingly, the analysis also supports the safety criterion of the system. Figure 9 .

Experimental Evaluation of the System Performance
This Section presents the experimental evaluation of the closed loop control of the 3 DoF planar manipulator. The aim of this experiment is to verify simulation results with the control performance of the manipulator in real-time. Three linearized DC actuators are controlled through the PID controller. The actual positions of each motor are measured by linear potentiometers moving parallel to the motor shaft as shown in Figure . The real-time control architecture is implemented on MATLAB SIMULINK environment. The tracking performance of each linear actuator affects the control of the end-effector. Accordingly, performance evaluation of each actuator is evaluated based on two different reference input signals. Figure A presents response of the actuator measured by the linear potentiometer when the motors are conditioned by step input. The experimental results indicate that each motor follows the given reference with tracking accuracy of 94.6%, 96.86%, and 96.8%. Moreover, the control performance of each motor is also evaluated based on the sine response. Accordingly, the position tracking accuracy of the first, second and third actuator is 89.8%, 88.8%, 84.3%, respectively. The small error was observed basically due to the friction in the linear potentiometer; but can be inhibited by improving the mechanical imperfection inside the tube. Along these lines, the robust controllers employed with high gains and control rates allow the tracking error of the actuators at low level. Figure 10 .

Evaluation of Balance Skills and Reaction Time Measurement
The pressure distribution under the soles of the feet depends on the balance skill of a person while performing the stability tasks. These tasks are aimed to detect the patient's pressure distribution in both static and dynamic conditions. Moreover, the other evaluation criterion for balance skill is to determine the amount of CoM deviation with respect to the ideal axes (center of the platform). Each platform constituting the end-effector is responsible for performing both assessments. Along these lines, in this Section, the evaluation of upper and lower platforms is carried out as means of 8 load cells positioned in accordance with the important pressure zones [46] under the sole of the human foot. The maximum load capacity of each load cell on the upper platform is 10 kg, and the capacity of those on the lower platform is 40 kg. The capacity of the load cells of the upper and lower platforms is determined based on the average pressure under the foot and the average human weight, respectively.
Considering the design criteria, the balance skill of a person should be assessed through the platform in both static and dynamic conditions. First of all, the capability of static balance evaluation of the platform is tested. In this experiment, three different cases of static balance of a person are considered. Given the load distribution of a foot, a water-filled jar is used to transfer the load less to the toe side than to the heel side as presented in Figure A. In the first case, a 160N plastic water-filled jar is placed on the left side of the upper platform to realize the standing position on the left foot (Case 1a) and only the load cell (lc)-1, lc-2, lc-3 and lc-4 are collected the data as depicted in Figure B. The same jar is replaced and put on the right foot slot to represent the standing position on the right foot (Case 1b), and the data due to the distributed load over the right side is gathered by the lc-5, lc-6, lc-7 and lc-8 as depicted in Figure B. In the second case, both jars are slightly tilted to the front of the platform, representing the state of inclining forward. As depicted in Figure C, the load distributed over the front side is measured mainly by lc-1, lc-2, lc-5 and lc-6; however, other load cells also measure the remaining load on them. In the last case, the opposite scenario of Case2 is implemented, that is, the jars are inclined to the back of the platform. Accordingly, the distributed load is mainly sensed by lc-3, lc-4, lc-7, and lc-8 as depicted in Figure D, but the rest of the load is also measured by other load cells. In this experiment, the sampling time of the implementation is 1ms and each case lasts 2 minutes. The mean of the collected data by each load cell is presented in Table 3.  Figure 11 . The evaluation of the upper platform in the dynamic conditions is carried out considering the four movements of the ankle joint, namely plantar flexion, dorsiflexion, inversion, and eversion movements. In the experiment, after placing the distributed loads as shown in Figure A as a foot model and waiting for calibration for 5 seconds, the manipulator follows the trajectory to realize,the standing straight condition, the plantar flexion, dorsiflexion, inversion, and eversion movements sequentially. Each movement is repeated by the parallel manipulator for 1 minute, and the data gathered from each load cell is sampled at a sampling time of 1 ms. The mean and standard deviation of the collected data for each load cell are presented with box plots in Figure A. The two plots in Figure A correspond to pressure distributions of the left and right feet, respectively. The experimental results indicated in Figure A present that the pressure distribution due to the water-filled water jar on the platform is directly correlated with the type of movement of the platform that mimics ankle movement. That being said, the load cells located on the upper platform are sensitive to the configuration of the end effector in space. In other words, the manipulator is tilted by means of PID controller as if the ankle behaves like the plantar flexion, thus the regions indicated by the numbers of 1, 2 and 5, 6 of the upper platform measure higher pressure relative to the remaining parts. As for dorsiflexion type motion of the platform, lateral regions, regions 3 and 7 have higher pressures. For the inversion type motion, the load causes more pressure mainly on regions 2, 3, 6 and 7, and slightly more on regions 4 and 8, whereas for the eversion type motion of the platform, more pressure is detected on regions 2, 3, 6 and 7 and slightly more on regions 4 and 8.
Lastly, the lower platform is also evaluated in the same experiment as both platforms are attached. During the experiment, the CoM of the water-filled jar is altered due to the different movements of the parallel manipulator as shown in Figure B. Each ring with different color in Figure B states the CoM of the water-filled jar in x-y coordinate system for various movement of the planar manipulator. Figure 12 .

Conclusion
In this study, kinematic and dynamic analyses, design, implementation and experimental evaluation of a 3-DoF robotic platform for the rehabilitation of balance skill and improvement of reaction time of MS patients have been presented. Simulation results and experimental evaluations indicate that the robotic platform is faithfully adapt to the ankle kinematics and constraints. The novel design of the end-effector composed of two platforms are also experimentally evaluated in both static and dynamic conditions. The results present that the proposed design successfully evaluates the balance skill of a person through both upper and lower platforms responsible for pressure distribution of feet and the CoM of the body, respectively. Video demonstrating the working principles of the robotic platform and its two-fold end-effector, also an illustrative example of the platform working in harmony with a task presented in a VR environment is available at https://www.youtube.com/watch?v=fmmZm8K6Szg . Please note that the first prototype presented in the paper is implemented only as a proof of concept, the overall system is not tested by human-subjects. In order to prove the design criteria of the robotic platform (as discussed in Design Objectives of 3 DoF Robotic Platform), a water-filled jar is used in the experimental evaluations of the end-effector. Such an experimental object allows us to realize the realistic scenario realized by the foot geometry, and accordingly, the pressure distribution of the object gradually increase from the toe to the heel, like that of a foot.
Our future study will focus on improving the capacity of both platforms endowed with the load cells to be able to realize extensive experiments with healthy volunteers and MS patients. Since the robotic platform can simultaneously evaluate the user's performance based on the measurements via upper and lower platforms and robot actuation force, such data gathered during the rehabilitation provide appropriate physical therapy orientation.  loads while the end-effector is in anthropomorphic motion S1 Video. Illustrative Representation Video demonstrates the working principles of both the robotic platform and its two-fold end-effector, also an illustrative example of the platform working in harmony with a task presented in a VR environment. The video is also available online at https://www.youtube.com/watch?v=fmmZm8K6Szg . S1 Data File. Data of Simulations and Experiments All data underlying the findings are provided with the article .mat format. This file does not contain identifying participant information. S1 Table. Foot-pad pressure Average foot-pad pressure ratios S2