Mechatronic design of a 3 degrees of freedom parallel kinematics manipulator with integrated force plate for human balance evaluation and rehabilitation

Mechatronic design of a 3 degrees of freedom parallel kinematics manipulator with integrated force plate for human balance evaluation and rehabilitation ✰ Francisco J. Campa * , Mikel Diez , Javier Corral, Erik Macho, Saioa Herrero, Charles Pinto Department of Mechanical Engineering, University of the Basque Country UPV/EHU, Escuela de Ingeniería de Bilbao, Plaza Ingeniero Torres Quevedo 1, Bilbao 48013, Spain ARTICLE INFO Associate Editor: Michael G. Ruppert Keywords: Human balance Dynamic posturography, Rehabilitation Parallel kinematics Force plate ABSTRACT One of the main sequelae of stroke is hemiparesis or partial hemiplegia. This condition causes patients to undergo lengthy rehabilitation processes to recover balance and gait. Thus, it is required to have tools to systematize rehabilitation tasks while monitoring the evolution of the patient over time. This paper presents a new prototype based on a 3PRS parallel kinematic manipulator with a force plate in the end effector to rehabilitate and evaluate balance by measuring the Center of Pressure of the patient, a widely studied parameter in the field of posturography. The prototype has been designed to be able to measure in a reliable and repeatable way and, as it is a medical device, it includes design considerations related to ease of use, visual feedback and safety. The paper describes the kinematics, dynamics, mechatronic modelling and mechanical design of the parallel kinematics machine, as well as the design of the force plate in charge of determining the location of the Center of Pressure. Also, a formulation is proposed to correct the influence of the inertial and gravitatory forces on the top plate during the motion of the platform on the measurement. Finally, the prototype has been evaluated experimentally to compare the motion with the developed models and to determine the precision and robustness against noise and motion related effects. Abbreviations TUG Timed Up and Go test SPPB Short Physical Performance Battery DGI Dynamic Gait Index BBS Berg Balance Scale COP Center of Pressure AP Anteroposterior ML Mediolateral DOF Degrees of Freedom PRS Prismatic-Revolute-Spherical joints PID Proportional-Integral-Derivative Controller FEM Finite Elements Method K zz ZZ term of the stiffness matrix of the FEM model V-M Von Mises equivalent stress SF Safety Factor OWR Operational Workspace of Rotation H G Angular momentum of the top plate J i Actuator inertia moment in motor axis c i Actuator viscous friction in motor axis τ Ci Actuator Coulomb friction in motor axis m i Mass of plate that links revolute joint and actuator m BC Manipulator bars mass I BC Manipulator bars inertia in their mass center m MP Mobile platform total mass I xG,yG,zG Mobile platform inertia moments in the mass center p Actuators pitch K v Proportional gain of the position controller K p Proportional gain of the velocity controller K i Integral gain of the velocity controller 1. Introduction Globally, stroke is the second leading cause of death and the third ✰This paper was recommended for publication by Associate Editor: Michael G. Ruppert * Corresponding author. E-mail address: [email protected] (F.J. Campa). Contents lists available at ScienceDirect Mechatronics journal homepage: www.elsevier.com/locate/mechatronics https://doi.org/10.1016/j.mechatronics.2024.103278 Received 8 March 2024; Received in revised form 24 October 2024; Accepted 20 November 2024 Mechatronics 105 (2025) 103278 Available online 25 November 2024 0957-4158/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license ( http://creativecommons.org/licenses/bync/4.0/ ). leading cause of disability [1]. Stroke is caused by the death of some brain cells due to lack of oxygen, usually caused by the loss of blood flow from either a blockage or rupture of an artery. The incidence of stroke has varied in recent decades, with the incidence doubling in developing countries and almost halving in developed countries [2]. On average, stroke is suffered 15 years earlier, mainly due to unhealthy habits in developed countries. Depending on the severity, stroke can result in everything from impaired language and/or motor skills, sensory alterations, to death [3]. Specifically, the loss of mobility due to total or partial hemiparesis, loss of strength or motor response of the upper or lower limbs of half of the body, is one of the most serious long-term effects. In [4] it is shown that 32 % of stroke patients have severe problems or are simply unable to walk, compared to about 9 % for all other conditions. This means that the most basic daily tasks are hampered by these mobility-related sequelae. Mobility is considered a crucial factor in healthy ageing as it generally determines a person’s ability to change position or to move around. When it comes to rehabilitating patients affected by neurological damage, it is essential to address initially the function of balance, then the gait and, finally or in parallel with the second, the transfer or ability to move an object from one position to another. It is therefore necessary to have tools that allow an accurate measurement of the degree of loss of balance and gait function, both for the initial diagnosis and to monitor the patient’s evolution during rehabilitation. A major obstacle in the diagnosis and treatment of balance is that it depends on the integration in the central nervous system of information received from three main sensory systems: vestibular, visual and somatosensory [5]. Nowadays, functional measurement tests are used to diagnose balance function, among which there are more than thirty protocols, including the TUG, the SPPB, the DGI or the BBS [6–9]. These tests differ in the type of exercises and the interpretation of the results, since, while some methods involve quantitative measurements of time or number of repetitions, others focus on qualitative aspects that are not always assessed by the same person. Also, none of them is suitable for all patients, and in most cases, they are not able to discriminate between patients from different clinical groups: age, state of the rehabilitation process or sensory system affected [10–12]. Hence, there is a need for devices capable of taking objective measurements to assess the patient and that are able to combine a variety of exercises, from standing upright on an inclined floor, to walking, or performing functional exercises that simulate everyday actions such as picking up a glass from a cupboard. On the other hand, it is widely accepted in the literature that the Center of Pressure (COP) is a parameter related to the state of the balance function [13,14].The COP is the point of application of the resultant force applied to the surface where the patient is standing, and in static conditions it should be aligned with the Center of Mass (COM) of the person, see Fig. 1, where the reaction on the feet is depicted and the total reaction that balances the weight is applied in the COP. However, it is still not clear which indicators deducted from the COP motion are the most relevant to diagnose balance. In this sense, different metrics obtained from the COP motion have been proposed in the literature [15–18]: maximum, minimum and mean COP displacement in the anteroposterior (AP) and mediolateral (ML) directions, shape of the confidence ellipse in a statokinesigram, maximum, minimum and mean COP velocities in AP and ML directions, and Fourier analysis of the COP motion. In [19] it is shown that some of these indicators show a good correlation with BBS and TUGT tests. In an attempt to find a solution to the problem of balance evaluation and rehabilitation, different equipment can already be found on the market. The Biodex Balance System® is a tilting platform system to train the balance function [20]. BalanceTutor from MediTouch is a moving treadmill platform with a moving belt that allows forward, backward and lateral movement for static and dynamic balance rehabilitation [21]. The Huber 360 is a multi-axial moving platform that incorporates force sensors in both the platform and the handgrips [22]. In [23,24] the design of a 2-degree-of-freedom platform for ankle and lower trunk rehabilitation is shown. The platform has a gimball-based design achieving rotation around horizontal axes by means of two motors, one of them not fixed to the fixed frame. This system measures the forces exerted by the patient on the platform and the movement of the patient with an IMU. Other commercial systems integrate not only the movement of the platform, but also virtual environments for patient training. For example, [25,26] show a 6 degrees of freedom (DOF) platform based on the Stewart architecture with a treadmill that allows the patient to walk while simulating different environments or perturbations. In terms of balance or gait analysis, it is a very complete solution as it integrates both the force sensors on the treadmill and the possibility to track the body using motion capture. Systems in a research stage can also be found in the literature. For example, [27] shows the design of a 3 DOF spherical platform that allows the platform on which the patient stands to be rotated around the three axes, allowing tilting movements. In the validation study, markers placed on the subjects were used to estimate their movements. One limitation of this system is that the use of markers would not be possible to implement in patients due to the high preparation time involved. Other prototype can be found in [28], where the design procedure of a 3PRS parallel mechanism is described for the evaluation of vertigo in patients. In this paper, a new parallel platform based on the 3PRS architecture is proposed for the assessment of balance in patients who have suffered a stroke. In comparison with the State of the Art, the platform, named OREKA, presents a much wider moving force plate and an easier and wider access to the system. The paper is organized as follows. In Section 2the design of the system is presented, from its requirements to the kinematics, dynamics and mechanical design. In Section 3 the experimental validation of the mechatronic system is presented, describing the force plate performance and its accuracy in Section 4. Finally, the main conclusions are presented in Section 5. 2. Mechatronic design 2.1. Design requirements The requirements were defined together with the medical staff from Hospital Gorliz form the basque public healthcare system to cover the Fig. 1. Center of Pressure (COP) and Center of Mass (COM) alignment in static conditions. F.J. Campa et al. Mechatronics 105 (2025) 103278 2 functional needs: ergonomics, user experience, range of motion and velocity, safety aspects and measurements to be carried out. Additionally, the UNE-EN ISO 12100, UNE-EN ISO 14971 and UNE-EN 60204-1 standards related to medical and electrical devices were consulted. Hence, the following basic design requirements were established: - Maximum tilt range of the platform of ±15◦in any direction with respect to the horizontal plane, with a maximum tilting velocity of 20 deg/s. - Patient weight up to 150 kg. - Precision of 2 mm in the measurement of the COP position. - Easy access to the device, large platform to do exercises in different postures: standing, tandem, semitandem. - Deformations lower than 1 mm in any moving part. - Safety measures as an emergency button, an entrapment detector, or a fall detector. 2.2. Kinematics of the manipulator The manipulator must introduce a disturbance on the force plate that stimulates the response of the patient. It was decided to introduce a vertical motion and a tilting around any horizontal axis. Hence, a 3PRS type parallel kinematics mechanism has been proposed [29]. In the moving platform there are three S-joints at points B i in a triangular arrangement of base 2R and height D, as shown in Fig. 2, with P being the origin of the mobile reference system UVW, with the U-axis aligned with PB 3 and the W axis perpendicular to the moving platform. The platform is connected to the linear actuators through three equal B i C i bars of length L. These are joined to the linear actuators through the R-joints at points C i , and translate vertically according to the variation of s i , coordinates of the joint space. The origin of s i coordinates is in points A i , which shape a triangle with base 2H and height N. The origin of the fixed reference frame XYZ is at point O, the direction of the X-axis coinciding with OA 3 and being Z-axis in vertical direction. 2.2.1. Parasitic motions The selected arrangement with bars B 1 C 1 and B 3 C 3 in the same plane minimizes parasitic motions in the platform [30] and allows a wider access attending to the requirements of the design. Indeed, the number of DOF of the platform are 3: vertical translation in Z-axis according to z P , rotation around the X-axis according to ψ , and rotation around the Y-axis according to angle θ. That means that the parasitic motions are the translations in X-axis and Y-axis, x P and y P respectively, and the rotation φ around Z-axis. After imposing the kinematic restrictions of the present arrangement, where S-joints B 1 and B 3 remain in the y =0 plane and S-joint B 2 in the x =0 plane, it is demonstrated that the displacement in the Y-axis in P and the rotation around Z are zero [30], see Eq.1. xP= − Dsin ψ sinθyP=0ϕ=0 (1) Consequently, the rotation matrix R that relates the moving reference system with the fixed one is: R=⎡ ⎣ cosθsin ψ sinθcos ψ sinθ 0 cos ψ −sin ψ −sinθsin ψ cosθcos ψ cosθ⎤ ⎦(2) 2.2.2. Inverse and direct kinematic problem Imposing the condition that the vectors C i B i have a constant length of L [31], the position of the actuators s i is obtained as a function of the position of the platform defined by z P , ψ and θ. s1=zP+Rsinθ+ L2− (Rcosθ−H+Dsin ψ sinθ)2 √ s2=zP+Dcosθsin ψ + L2− (N−Dcos ψ )2 √ s3=zP−Rsinθ+ L2− (H−Rcosθ+Dsin ψ sinθ)2 √ (3) On the other hand, the position of the platform is calculated as a Fig. 2. Kinematic model of the manipulator decoupling the mechanism from the actuators. F.J. Campa et al. Mechatronics 105 (2025) 103278 3 function of the actuators position s i , solving the following system of three nonlinear equations, one per bar, using the Newton–Raphson method where the initial guess is the default position at zero tilting: ( − Dsin ψ sinθ−Rcosθ+H)2+ (zP+Rsinθ−s1)2=L2 (Dcos ψ −N)2+ (zP+Dcosθsin ψ −s2)2=L2 ( − Dsin ψ sinθ+Rcosθ−H)2+ (zP−Rsinθ−s3)2=L2 (4) 2.2.3. Mechanism synthesis To achieve the desired workspace with rotations up to ±15◦, a synthesis procedure has been carried out following the one described in [31]. It has been considered that the moving platform where the patient is going to be positioned, must have a minimum width of 500 mm and length of 600 mm to allow several standing poses as required. Attending to these requirements, the following dimensions are achieved: Table 1 2.3. Dynamics and control modelling 2.3.1. Inverse dynamic problem To be able to compute the required forces to move the manipulator, the inverse dynamic problem of the manipulator has been solved, following the guidelines of [32], where the use of the Principle of energy equivalence is proposed. Instead of studying the dynamics of the complete parallel mechanism, which is a complex task due to the constraints of closed kinematic chains, the Principle of energy equivalence states that the mechanical energy of each of the n subsystems in which the manipulator can be disassembled is equal to the energy of the complete manipulator, provided that they move as if they were assembled. The subsystems selected for the study of this mechanism have been the mobile platform, the three bars and the three actuators. This method allows analysing the dynamics of each subsystem in a simpler way. Thus, for a subsystem j whose generalized coordinates are q j , the equations of motion are obtained from the Lagrange’s equations: fj=d dt ∂ Lj ∂ ˙ qj − ∂ Lj ∂ qj (5) where f j is the generalized force on the element. The condition that all subsystems move as the assembled mechanism implies that q j is a function of the generalized coordinates of the assembled mechanism q s . Therefore, the virtual displacements δq j are related to the virtual displacement δq s by the corresponding Jacobians: δqj= ∂ qj ∂ qs δqs=Jjδqs(6) Also, the virtual work in the assembled mechanism is equal to the total of the virtual works in all the subsystems. The virtual work of the joint forces is not computed as it is cancelled with the one in the adjacent element. δWs=δqT sfs=∑ n j=1 δWj=∑ n j=1 δqT jfj(7) Substituting Eq. (6) in Eq.7, the forces f s to apply on joints B i to move the mechanism are: fs=⎧ ⎨ ⎩ F1 F2 F3⎫ ⎬ ⎭ =∑ n j=1 JT jfj(8) 2.3.2. Control To model the global dynamic behaviour of the manipulator and the control, the approach proposed by [33] has been followed. The dynamics of the manipulator are decoupled from the dynamics of the actuators, in such a way that the forces F i on the actuator table C i to move the manipulator are perceived as a disturbance F id by the actuators, as it is shown in Fig. 2. The position control of the manipulator is based on a joint-space control approach, as it is shown in the Simulink model in Fig. 3. The position command in workspace coordinates (z c , ψ c , θ c ) is fed through the inverse kinematic problem to calculate the position commands in the joint space (s 10 , s 20 , s 30 ), which are controlled using a cascaded PID control of position, velocity and current. Then, the direct kinematic problem is applied to calculate the end effector position (z s , ψ s , θ s ) from the actuators simulated position (s 1 s , s 2 s , s 3 s ). Regarding the modelling of the position control of the actuators in Fig. 4, a P controller is used in the position loop, and a PI in the velocity and current loops. This control approach was selected as a compromise between a low tracking error and a good response against external disturbances, which will come mainly from the weight of the patient above the force plate apart from the friction in the mechanical components. The linear position command s i0 is converted to angular position through the variable s2th =2 π /p considering the lead p of the ball screw. K v , K p and K i are respectively, the proportional gain of the position control, and the proportional and integral gains of the motor velocity control. As the current loop runs faster, the conversion from current to torque is assumed to be immediate through the motor torque constant K t , thus simplifying the current loop. A 1 DOF rotational model has been assumed to model each i-th actuator dynamics, considering only the total inertia, J aci in Fig. 4. The disturbances are the force F id that mechanism does on the actuator, which is converted to disturbance torque considering the ball screw lead p through the variable th2s =p/(2 π ), the weight of the actuator table MG aci , and the Coulomb and viscous friction, which were initially estimated and then identified experimentally once the prototype was built. Finally, the simulated angular position of the motor is converted again to linear position s is making use of the variable th2s. 2.4. Mechanical design After considering other requirements such as weight, total volume or maximum operating height, the design of the parallel mechanism converged to the model presented in Fig. 5. The mobile platform consists of two 7075-T6 aluminum plates measuring 800 ×800 mm, with a minimum thickness of 12 mm and reinforcement ribs with a maximum thickness of 20 mm to increase its bending stiffness, as shown in Fig. 6. The upper plate is the one on which the patient stands, and the lower one is articulated to the bars. Four uniaxial force sensors link the plates, in such a way that the weight of the top plate and the patient is transmitted to the bottom plate through the sensors. The validation of the design in terms of strength and stiffness is shown in the following subsections. 2.4.1. Strength validations To validate self-designed parts, a FEM model of the whole mechanism in Fig. 5 was developed. Every link of the kinematic chain was assembled with the corresponding joint to ensure a realistic force transmission, see red lines in Fig. 7. Starting from the CAD geometry, a simplified version without chamfers or bolt holes was developed, so a simpler mesh is obtained, improving the computational cost. The two plates of the mobile platform are joined by the four force sensors, which are substituted by beams in the model. The simplified FEM model consists of a total of 24 different parts without considering the fixed element and actuators, which are replaced by boundary conditions. These parts are connected by 23 kinematic conditions that represent the joints. For the static study, after several initial iterations it has been shown that the wedges and the spherical Table 1 Dimensions of the kinematic model of the platform. L (mm) R (mm) D (mm) H (mm) N (mm) 400 350 600 450 700 F.J. Campa et al. Mechatronics 105 (2025) 103278 4 joint hardly undergo any deformation, so as it can be seen in Table 2, they will be considered as rigid solids. A sensitivity analysis of the mesh has been performed to select the element size in each part to obtain a good computational accuracy. The resulting model consists of 230,000 higher order tetrahedron elements. The average computation time for each analysis position is of the order of 2 min on a computation station with 4 cores working in parallel and 32 Gb of RAM memory. Fig. 3. Mechatronic model of the manipulator. Fig. 4. Dynamic model of the i =1 actuator. Fig. 5. OREKA platform CAD model. Fig. 6. Location of the sensors in the mobile platform and details of the ribs. F.J. Campa et al. Mechatronics 105 (2025) 103278 5 Strength simulations have been conducted considering a 1.5 kN vertical force in the most critical orientation of the mechanism. For a safe-side design, the application point of the force has been changed along the top plate until the critical position for each element was achieved. As both the materials used in the prototype are considered ductile, Von Mises equivalent stress (V-M stress) has been calculated. Considering the strength of the respective material, the safety factor (SF) of each part was calculated to validate the strength of the prototype, see Table 2. All the safety factors are bigger than unity, validating then the prototype. In fact, the highest safety factors in some parts are due to the deformation requirement in Section 2.1. 2.4.2. Statics: vertical stiffness It is well-known that the stiffness of parallel manipulators varies over the workspace. To ensure that the stiffness design requirement is met, an analysis of the stiffness will be carried out by means of a FEM model. For the study, only the elements of the stiffness matrix involving the vertical load component F z will be considered since the origin of the applied forces comes from the weight of the patients. Furthermore, the stiffness analysis is going to be limited to K zz as reported during the requirements stage. When patients step onto the platform to perform the balance tests, this stiffness component has special significance in terms of confidence. Both the base frame and the travel guides of the drives have a very high stiffness, so that the analysis of the structural behaviour in the working space can be carried out independently of the Z coordinate at which the manipulator is positioned. The study shall consider only the possible dependence of the structural behaviour of the manipulator on the orientations. Likewise, the operational workspace of rotation (OWR) is defined by the maximum and minimum orientations that the manipulator can reach to fulfil its function without interference between its parts, (-15◦, +15◦) in the case of the present 3PRS manipulator. Consequently, for the study of the structural behaviour proposed here, the operating range of rotation in each of the orientations has been discretised in intervals of 3◦, resulting in 11 values for each rotation. This makes a total of 121 analysis points. Fig. 8 shows K zz variation over the OWR, where K zz value is represented in the z axis as well as with the colorbar. Also, the most representative values of K zz are presented in Table 3: For the K zz component, the maximum value is at location (3◦,15◦), while the minimum is at location (-15◦,15◦). From the values in Table 3, the highest value is twice as high as the lowest value and, in addition, the mean and standard deviation indicate that the stiffness values across the map are high. However, as can be seen in Fig. 8, the stiffness has a maximum for a pitch of ψ =3◦and decreases more sharply when decreasing ψ than when increasing it. Another noteworthy aspect is that the minimum of the function occurs for a value of ψ =-15◦where, in addition, it presents a maximum sensitivity to the variation of the parameter θ. Evaluation exercises for stroke patients will comprise independent orientation variations. Mainly two variations, θ varying between (-13◦,6◦) while ψ remains zero and, then θ constant and ψ varying up to maximums of (-8◦,8◦). For the most common trajectories foreseen for the use of the machine in equilibrium treatment and diagnosis, see second row in Table 3, the stiffness is high and uniform during the rotations associated with the most common movements. For a patient of about 150 kg, as required, the platform would undergo displacements smaller than 1 mm in most of the exercises. This result verifies the imposed requisite in the design stage. 2.5. Force plate design and COP measurement To measure the position of the patient’s COP during the tests, the force plate is composed by the two aluminium plates measuring 800 × 800 mm and the four uniaxial force sensors. The sensors measure the force in the direction perpendicular to the platform surface, so that by combining their values it is possible to deduce the position of the COP. The sensors arrangement in the force plate can be seen in Fig. 6. To determine where the COP is located on the top plate surface, a mobile reference system X’Y’Z’ is located on the top plate surface, in the center O’ of the rectangle defined by the sensors located in S i . Its dimensions are 2a x 2b, where a =250 mm and b =300 mm. It must be taken into account that the platform will be tilted, and the inertial forces and weight on the top plate will be measured also, apart from the weight of the patient. Assuming that mg and m are respectively the weight and the mas of Fig. 7. Symbolic representation of the kinematic constraints of the FEM model. Table 2 Number of parts and materials for FEM model and their performance. (*) For commercial elements. Part Material Elastic behaviour V-M stress (MPa) Strength (MPa) SF Upper plate 7075-T6 Aluminium Elastic 39.9 500 12.53 Sensor (*) AISI 1045 Steel Beam As per datasheet instructions Lower plate 7075-T6 Aluminium Elastic 2.4 500 208.33 Wedge 7075-T6 Aluminium Rigid 1.11 450 405.41 Spherical joint (*) AISI 1045 Steel Rigid As per datasheet instructions T-link AISI 1045 Steel Elastic 201.75 450 2.23 Ball bearing unit (*) AISI 1045 Steel Elastic As per datasheet instructions Support plate 7075-T6 Aluminium Elastic 36.62 500 13.65 Fig. 8. Kzz stiffness map in the OWR. F.J. Campa et al. Mechatronics 105 (2025) 103278 6 the top plate and w is the resultant vertical force received in the contact with the patient, their projection over the plate surface in a tilted position is calculated using the transposed rotation matrix R T in Eq. (2). In static conditions and with the plate horizontal, w should be equal to the weight of the patient, but in a general position it is expressed as follows. w=⎧ ⎨ ⎩ wxʹ wyʹ wzʹ⎫ ⎬ ⎭ =RT⎧ ⎨ ⎩ 0 0 −w⎫ ⎬ ⎭ =⎧ ⎨ ⎩ wsinθ −wsin ψ cosθ −wcos ψ cosθ⎫ ⎬ ⎭ (9) Hence, studying the translational dynamics of the top plate in Z’ direction, perpendicular to the plate, Eq. (10) is obtained, where f iz are the forces measured by the sensors and a Gz’ is the Z’ component of the acceleration of the mass center of the top plate. ∑ 4 i=1 fizʹ− (mg +w)cos ψ cosθ=maGzʹ(10) The acceleration of the mass center of the top plate is calculated as follows as a function of the acceleration of P, the origin of the mobile reference system UVW in Fig. 2, the angular velocity ω and angular acceleration α of the plate and the relative position vector PG between the mass center and P: aG=aP+ α ×PG + ω × ( ω ×PG)(11) To obtain a P , first, the position of P in the global frame XYZ is: OP =⎧ ⎨ ⎩ −Dsin ψ sinθ 0 z⎫ ⎬ ⎭ (12) Differentiating: d2OP dt2=⎡ ⎢ ⎢ ⎣ cos ψ sinθsin ψ cosθ0 0 0 0 0 0 1 ⎤ ⎥ ⎥ ⎦⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨ ψ ¨ θ ¨ z ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ +... ...⎡ ⎢ ⎢ ⎣ −sin ψ sinθ2cos ψ cosθ 0 0 0 0 ⎤ ⎥ ⎥ ⎦{˙ ψ 2+ ˙ θ2 ˙ ψ ˙ θ} (13) Applying the transposed rotation matrix, the acceleration of P in the moving frame is: aP=RTd2OP dt2(14) Finally, the angular velocity and acceleration in the moving frame are: ω =⎧ ⎨ ⎩ ˙ ψ ˙ θcos ψ − ˙ θsin ψ ⎫ ⎬ ⎭ α =⎧ ⎨ ⎩ ¨ ψ ¨ θcos ψ − ˙ θ˙ ψ sin ψ −(¨ θsin ψ + ˙ θ˙ ψ cos ψ )⎫ ⎬ ⎭ (15) From Eq. (10), the force w received from the patient is: w=∑ 4 i=1fiz −mgcos ψ cosθ−maGzʹ cos ψ cosθ(16) Regarding the rotational dynamics of the top plate, the theorem of the angular momentum is applied in X’ and Y’. Taking into account the symmetry in Y’Z’ of the plate, and the fact that the center of gravity G of the plate is in that plane, see Table 4, the angular momentum H G is: HG= { e1e2e3}⎡ ⎢ ⎢ ⎣ IxʹG0 0 0IyʹG−CxʹG 0−CxʹGIzʹG ⎤ ⎥ ⎥ ⎦⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ω xʹ ω yʹ ω zʹ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ =... ... =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ IxʹG ω xʹ IyʹG ω yʹ−CxʹG ω zʹ −CxʹG ω yʹ+IzʹG ω zʹ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (17) where: e 1 , e 2 , e 3 are the unit vectors of the moving frame; I x’G , I y’G, I z’G are the moments of inertia with respect to the axis of a frame with origin in G and parallel to the moving frame; C x’G is the product of inertia with respect to the two planes of that frame that intersect in the x G’ axis. Upon differentiation: dHG dt =⎧ ⎨ ⎩ IxʹG α xʹ IyʹG α yʹ−CxʹG α zʹ −CxʹG α yʹ+IzʹG α zʹ⎫ ⎬ ⎭ + ω ×HG(18) The force w is applied on the COP, point Q, and creates a moment with respect to the mass center G that is expressed as: NGW=GQ ×w=⎧ ⎨ ⎩ GQyʹwzʹ−GQzʹwyʹ GQzʹwxʹ−GQxʹwzʹ GQxʹwyʹ−GQyʹwxʹ⎫ ⎬ ⎭ (19) Considering that the COP will be on the top plate surface, z’ Q =0, and that the distance O’G x’ is zero, as shown in Table 4, the vector GQ is: GQ = − OʹG+OʹQ=⎧ ⎨ ⎩ xʹQ −OʹGyʹ+yʹQ −OʹGzʹ⎫ ⎬ ⎭ (20) The top plate lies on the four sensors, which impose a restriction to the displacement in X’ and Y’ and measure the force in Z’. Hence, the moment of those forces with respect to the mass center is: NGF=∑ 4 i=1 GSi×fi=∑ 4 i=1⎧ ⎨ ⎩ GSiyʹfizʹ−GSizʹfiyʹ GSizʹfixʹ−GSixʹfizʹ GSixʹfiyʹ−GSiyʹfixʹ⎫ ⎬ ⎭ (21) Hence, the theorem of the angular momentum in X’ and Y’ results in the following equations: xʹ)(−OʹGyʹ+yʹQ)wzʹ+OʹGzʹwyʹ+... ... ∑ 4 i=1(GSiyʹfizʹ−GSizʹfiyʹ)=(dHG dt )xʹ yʹ) − OʹGzʹwxʹ−xʹQwzʹ+... ... ∑ 4 i=1(GSizʹfixʹ−GSixʹfizʹ)=(dHG dt )yʹ (22) In Eq. (22), the terms dependent on the forces fix’ and fiy’ are neglected because they are not measured by the uniaxial force sensors. This is a simplification that introduces an error in the measurement, however, they are relatively small terms due to the small tilt of the table Table 3 Characteristic values of K zz in the OWR. K zz (N/mm) Min. Max. Av. Dev. -15◦< ψ <15◦, -15◦<θ <15◦1.59⋅10 3 3.12⋅10 3 2.80⋅10 3 283.96 -13◦< ψ <6◦, -8◦<θ <8◦ 2.93⋅10 3 3.04⋅10 3 2.96⋅10 3 30 Table 4 Inertial and geometrical properties of the force plate. i-th sensor i =1i =2i =3i =4 GS ix’ (mm) -250 -250 250 250 GS iy’ (mm) -317.15 282.85 282.85 −317.15 GS iz’ (mm) -1.69 O’G (mm) O’G x’ O’G y’ O’G z’ 017.15 -7.31 I axisG (kgm 2 )I x’G I y’G I z’G 0.4758 0.4639 0.939 C x’G (kgm 2 ) -0.0001943 F.J. Campa et al. Mechatronics 105 (2025) 103278 7 and the fact that they are multiplied by GSiz’, two orders of magnitude smaller than GSiz’ and GSiz’, as seen in Table 4. As a result, the position of the COP depends on the platform mass center position, the inertial term due to the plate motion, and the forces measured. Substituting Eq. (9) into Eq. (22), the coordinates of the COP, x’ Q and y’ Q , are calculated as a function of the platform position as: xʹQ=OʹGzʹtanθ cos ψ +... ... 1 wcos ψ cosθ[(dHG dt )yʹ +∑ 4 i=1(GSixʹfizʹ)] yʹQ=OʹGyʹ−OʹGzʹtan ψ −... 1 wcos ψ cosθ[(dHG dt )xʹ −∑ 4 i=1(GSiyʹfizʹ)] (23) 2.6. Description of the prototype Once the mechanical design of all the elements has been completed, the commercial elements are selected. Three 3.6 kW synchronous motors from the manufacturer B&R model 8LSA55.DA030S000-3 are selected, which are controlled by ACOPOS P3 controllers, model 8EI013HWS10.0200-1. Each motor has a safety brake that is activated in the event of a loss of power to the machine. These motors are connected to ball bearing linear guides model EGC-BS-KF-GK from FESTO with a pitch of 40 mm/rev. The motor-guide assembly completes the vertical actuator required to drive the mechanism. Each vertical actuator has a maximum linear speed of 2 m/s, and a maximum continuous force of 1800 N, i.e. they have been oversized so that each one separately could support up to 180 kg of weight. The rods connecting the platform and the linear guides are connected to the platform by Hephaist SRJ024C spherical joints, and to the linear guides by SKF SY20TR ball bearing revolution joints. Motion control is provided by the controller on the B&R panel PC 3100. This control is closed, i.e. the user of the machine is only able to program a series of predetermined movements, agreed as suitable between the UPV/EHU team and Hospital Gorliz. The panel PC, motor controllers and main electrical components are installed in a separate cabinet so the medical staff can use it more comfortably. As explained in Section 2.5, the mobile platform is a force plate and is responsible for taking measurements to determine the position of the COP during the exercises programmed in the machine. The four sensors installed in the platform are Interface GWMC based on strain gauges with a maximum range of up to 2000 N. The sensors signals are amplified by Interface SGA amplifiers, integrated into the machine structure. On the other hand, three sensorized hand rests have been installed. One is frontal, and two are lateral at about 45◦, so that the patient can hold on if necessary or as indicated in the exercise. Each of the supports has three uniaxial force sensors based on strain gauges of 250 N range model TE ELAF-T1 M, that reflect the extent to which the patient leans on the hands to maintain balance during the exercise. The data collected by all the sensors is provided to the rehabilitation doctor via USB, who has a software for processing them, accessing quantitative parameters about the evolution of the patient’s performance during the rehabilitation phase. Finally, as can be seen in Fig. 9, the mechanism is supported by a rigid structure formed by three levels of 4 and 8 mm horizontal steel plates joined by 4 mm thick steel braces in St. Andrew’s Cross. Thus, the machine dimensions are 1300 ×970 ×1400 mm with a total weight of 300 kg, which guarantee the stability of the assembly. The whole assembly is protected by an aluminium fairing which ensures that the patient cannot come into contact with any of the electrical elements of the system. To ensure the safety of the patient, additional security sensors have been installed. In this way, the perimeter of the moving platform is surrounded by a physical barrier on which photoelectric barrier sensors (in blue in Fig. 10) have been installed, so that if the patient, for any reason, inserts a limb between the platform and the fairing, the beam is interrupted, and the system stops without any harm to the patient. 3. Experimental validation of the manipulator dynamics The mechatronic model used in the design has been validated to check if it can predict the dynamic behaviour of the manipulator. The parameters used are in Table 5. First, the inertial parameters have been obtained from the CAD model and manufacturer datasheets: J i is the motors inertia, m i is the mass of the couplings between the actuators and the mechanism, m BC and I BC are the mass of the bars and their inertia in their center of gravity, m MP is the mobile platform total mass and I x’G, I y’G, I z’G the inertia of the mobile platform in the center of gravity in axis X’, Y’ and Z’. Also, the viscous c i and Coulomb friction τ Ci of the motors have been identified experimentally by running a series of tests at different constant angular velocities with the motors in isolation. Relating the torque and speed, the trend is fitted to a linear relation where the slope is the viscous friction and the value at zero speed is the Coulomb friction. Finally, the ball screw lead p and the control gains used, which are the same for the three actuators, are shown. Several tests have been conducted reproducing the exercises proposed by the medical staff with no person on the force plate. They are periodic rotations between two angular positions of the table. Here, the results for two exercises are shown: tilting only in antero-posterior (AP, around X axis) direction from -12.5◦to 6.4◦and tilting in medio-lateral (ML, around Y axis) direction from -8◦to 8◦, both with a trapezoidal Fig. 9. Schematic representation of the prototype. F.J. Campa et al. Mechatronics 105 (2025) 103278 8 velocity profile. All the experimental signals have been sampled at 50 Hz, and low pass filtered at 10 Hz. Two comparisons between the model and the experiments are shown: in Fig. 11 the torque of the motors and in Fig. 12 the tracking error in linear coordinates s i , that is, converting the angular error measured in the motor encoder to a linear error applying the lead p of the ball screw. The results obtained in both motions by the simulation are similar to the measurements, considering all the simplifications made and the fact that the dynamic parameters used are the theoretical ones. In fact, only friction in the motors has been experimentally identified, neglecting the friction in the joints, or the variations in mass or inertia due to geometrical deviations during the manufacturing of the components. Regarding the torque, the highest errors appear clearly in the second actuator, where an average error up to 0.5 Nm appears. As for the tracking error, there are peaks of error up to 1 mm in the measured position that are related to a small vibration that appears in the manipulator in the inversions of the movement. It is due to the inertial forces exciting the first modal frequency of the manipulator around 12 Hz, but they are not dangerous for the patient from the clinical point of view. In general, the behaviour of the manipulator in isolation is in line with the model prediction, but the results could be improved applying a dynamic parameter identification instead of using the theoretical dynamic parameters [34]. 4. Experimental evaluation of the force plate performance 4.1. Influence of noise and motion related forces The force plate was evaluated first in terms of noise rejection in static and dynamic conditions, that is, during motion. Regarding the noise, as the force signal measured in the sensors are subject to electromagnetic interference, the measured position of the Center of Pressure presents oscillations even with no load above the plate. Under dynamic conditions, the motion of the platform itself also introduces inertial and gravitational effects that are measured by the force sensors and must be compensated as shown in Section 2.5. The experimental methodology has been the following. First, the COP coordinates were measured during the execution of the exercises shown in Table 6. These represent the opposite extremes in terms of amplitude and velocity of motion from a total of 25 pre-programmed exercises available. Then, to evaluate the influence of noise and motion related effects, the standard deviation of the COP coordinates in each test was calculated. It was expected that the signal-to-noise ratio would depend on the load applied on the plate, the higher the force measured, the lesser the influence of noise. To prove that, in static conditions, 6 combinations of calibrated masses from 5.2 to 41.8 kg were located on the platform and measured for 30 s. To locate the masses a fixture plate with holes every 25 mm with a tolerance of 0.1 mm was used. The fixture plate was centred in the rectangle formed by the sensors with the aid of a custom fixture designed to place it using the contour of the top plate as a reference, as it is shown in Fig. 13. All the masses were machined with a lug in the center of one side to fix them in the holes of the fixture plate. The result is shown as Ex. 0 in Fig. 14 following the nomenclature of Table 6. The figure, in logarithmic axis, represents the standard deviation of the COP coordinates as a function of the mass located on the platform for the conditions in Table 6. With a 5.2 kg mass, the standard deviation reaches 0.72 mm for the y coordinate of the COP and falls to 0.08 mm for both coordinates with 41.8 kg. Hence, the influence of noise in static conditions is below 0.1 mm for a weight that could be considered as low for an adult patient and it will be even lower for weightier patients. In dynamic conditions, the calibrated masses were not used as the height of their mass center combined with the tilting of the table introduced a motion in the COP that did not allow isolating the influence of the noise and the motion. So, all the measurements were done under the conditions in Table 6 with no mass above the table and, instead, a virtual mass located in the center of the table was simulated adding ¼ of the virtual weight to the force signal of each sensor. In Fig. 14 the standard deviation in the measured COP coordinates is Fig. 10. Prototype built and installed in Hospital Gorliz. Table 5 Inertial and geometrical properties of the elements of the manipulator. PID controller gains. i-th actuator i =1i =2i =3 J i (kg⋅m 2 )0.0025 m i (kg) 1.188 m BC (kg) 1.186 I BC (kg⋅m 2 )0.025 m MP (kg) 52.31 I x’G,y’G,z’G (kg⋅m 2 ) I x’G =2.539 I y’G =2.51 I z’G =5.034 c i (Nm⋅s/rad) 0.00325 0.00155 0.00125 τ Ci (Nm) 0.49 0.5 0.49 MG aci , (N) 62.8 73.82 87.9 p (m/rev) 0.04 K v (1/s) 50 K p (A⋅s/rad) 2 K i (A/rad) 40 F.J. Campa et al. Mechatronics 105 (2025) 103278 9