Abstract

The objective of the present work is to develop a device for training the trunk balance and motion during the early stage of rehabilitation of patients who have suffered a stroke. It is coupled to a standing frame and is based on a parallel continuum manipulator where a wearable jacket is moved by four flexible limbs actuated by rotary motors, achieving the translation and rotation required in the trunk to perform a given exercise. The flexible limbs act as a natural mechanical filter in such a way that a smooth physiological motion is achieved, and it feels less intimidating to the patient. After measuring the kinematic requirements, a model has been developed to design the system. A prototype has been built and a preliminary experimental validation has been done where the jacket generates translation coupled to a rotation around the anteroposterior, medio-lateral and longitudinal axis. The measurements of the motors torque and the force sensors located in the flexible limbs have been compared with the simulations from the model. The results prove that the prototype can accomplish the motions required for the rehabilitation task, although further work is still required to control the interaction with the patient and improve the performance of the device.

Introduction

The main objective of physical rehabilitation is to improve the functional ability and quality of life of the patients. Nowadays, the aging in population has caused an increase of age-related diseases that affect the nervous system [1,2], thus increasing the need for treatment. Among them, stroke which is caused by a blockage of an artery or cerebral bleeding, has become an important cause of death or disability with a mortality rate of 11% [3], representing the first cause of death in women and the second in men in some countries [4].

The vast range of consequences of stroke means that the rehabilitation therapy must be specifically designed for every patient. However, one of the most common sequels is hemiplegia, where the patient experiences a loss of control on one side of the body, affecting the balance and gait. In these cases, as a previous stage to be able to stand up or walk, the rehabilitation starts focusing on the trunk balance training by a series of active-assisted exercises based on a high number of repetitions on a standing frame.

It is in these repetitive tasks where robotics has found a niche in rehabilitation. During the last years robotic systems have suffered a growth in healthcare applications, both in surgery to perform accurate tasks [5], and in rehabilitation doing autonomous exercises that complement the work of the physiotherapists [6]. In conventional physical rehabilitation, the fatigue of the physiotherapist caused by the high number of repetitions and the limited human resources limit the amount of sessions that the patient receives, slowing down the recovery [7]. A robot assisted therapy overcomes that problem, shortening the rehabilitation period also because of the use of sensors that collect quantitative and objective data about the patient's performance [8,9].

Most of the rehabilitation robots currently designed are dedicated to the treatment of lower and upper limbs. One of the first, the MIT-MANUS [10], had 2 degrees-of-freedom and an impedance control to perform elbow and shoulder rehabilitation on a horizontal plane. It proved that patients who had worked with the robot obtained a significant improvement in limb functionality [11,12]. The ARM-GUIDE is another device for upper limb rehabilitation with four degrees-of-freedom, where the patient's arm performs motions in a larger workspace [13]. The MIME system was based on a modified industrial PUMA robot [14] for mirror therapy of upper limbs. Regarding lower extremities rehabilitation, Lokomat consists of a treadmill where a robotic orthosis guides the patient in gait training, assessing the process with several sensors placed on the system [15].

Regarding trunk rehabilitation, most of the robots are focused on sit-to-stand rehabilitation where the upper and lower limbs are moved to achieve passive trunk rehabilitation [16]. In other words, trunk muscles can be strengthened through the force applied in upper or lower limbs. In this regard, in the work of Kamnik and Bajd [17], a three degrees-of-freedom mechanism is proposed for standing-up rehabilitation. The system collects quantitative data through infrared markers and a group of sensors placed on the prototype. Chapman et al. [18] present a novel clinical research to improve the rehabilitation process in postsurgery shoulder by collecting data of torso and shoulder mobility, carrying out an experimental identification of the motion characteristics of the trunk. Johnson [19] studies the effect of standing-up rehabilitation on the human trunk and highlights the relevance of using a multibody approach to analyze the mobility. Crosbie et al. [20] analyze the movement of the lower thoracic and lumbar spinal segments and the pelvis in gait exercises. Cafolla et al. designed a sensorized waistcoat to characterize the movement of the human trunk, applying then the results obtained to the development of humanoid robots [21,22]. Nevertheless, little work has been found in active rehabilitation to restore trunk mobility, which is an essential task to balance the body in seated, standing or walking situations.

On the other hand, the development of continuum mechanisms has suffered a similar progression as medical robotics. Instead of using traditional kinematic joints with rigid components, these mechanisms create motion through the deflection of flexible elements [23]. Due to their slender shape, their application to laparoscopic robotics has been immediate [24]. On the other hand, they are complex to design and control [25], as their behavior is nonlinear due to the large deformations of their components, and the selection of the optimal material for the slender elements becomes essential.

The present work presents the modeling, design and testing of a continuum parallel manipulator for rehabilitation of the trunk motion in hemiplegic patients. The prototype attempts to reproduce common rehabilitation exercises collecting data during them. First, the design concept is shown. Second, the measurement of the kinematic requirements is done. Third, the fundamentals of the cosimulation of patient and manipulator developed are explained. Then, the mechatronic design is shown, to end with the experimental testing carried out to validate the models and the prototype.

Design Concept

The aim of this work is to provide a solution for the rehabilitation of trunk in stroke patients who have lost the proprioception in half the body. One of the first steps in the rehabilitation is to recover the ability of balancing the trunk, as a previous step to train the balance in standing position and the gait. This stage takes place in a standing frame, which fixes the legs and hip of the patient, so it is possible to perform specific exercises for the trunk motion and balance training.

The rehabilitation device here proposed is based on a parallel continuum manipulator, see Fig. 1. The flexibility of the slender elements on these mechanisms brings several advantages: they are easier to wear, they feel less intimidating as the patient is not fixed to a rigid mechanism and has some freedom to move, they can absorb external disturbances and, finally, the motion achieved is smooth, “physiological” in physiotherapist's words, compared to the motion of an exoskeleton with rigid members, and this is because they naturally act as a mechanical filter for the motion created in the actuators.

The device here proposed consists on a jacket, which is the mobile platform of the parallel mechanism, and four flexible limbs linked in one end to the jacket by means of spherical joints (S-joints) and in the other end to the rotary motors that act as a revolute joint (R-joints). The patient keeps a standing pose up to the hip thanks to the standing frame that fixes the patients knees and hip. The motors, which are fixed to the standing frame, actuate the limbs that deflect thus generating translational and rotational motion in the jacket. The contact surface between patient and jacket is located on his back. The patient wears the jacket by means of several adjustable straps like those of backpacks, keeping a proper contact between patient and jacket, so the motion is properly transmitted.

The transmission of the motors rotary motion to the jacket or platform is achieved thanks to the deflection of the limbs. These limbs are composed by a coupling to the motor axis, a set of four backbones made from Nylon 6.6, a coupling where the bars are fixed, an uniaxial force sensor and another coupling to the S-joint, see Fig. 1. The central backbone is fixed in both ends, while the outer backbones can slide through the R-joint coupling, so they remain as parallel as possible to the central one while the orientation between couplings change. The force sensors have been introduced in the prototype to be able to check the normal forces in the limbs and compare them to the models developed during the design stage.

Kinematic Requirements Measurement

To obtain the ideal workspace of the prototype proposed, a series of experimental tests have been carried out measuring the motion of a person doing rehabilitation exercises indicated for stroke patients. A 26 years old healthy person, 72 kg weight and 1.79 m height, has performed the series of rotations of the trunk around the three axes of the human body.

The layout of the experimental tests is shown in Fig. 2. Two plates with Bertec triaxial sensors were used to capture the reaction forces and a 10 camera Vicon Systems vision system was used to record the movements during the test. In addition, three perimetral rings with eight reflective markers have been placed along the trunk. The markers of the upper ring were located on the internal and external scapular border and two on the major pectoral. In the intermediate ring, the markers were located on rectus abdominis muscle, oblique major of the abdomen and two on the dorso-lateral muscle. The markers of the lower ring were located on iliac crest, oblique abdominals and two in the erector spinae. Finally, two additional markers have been placed on the supra-sternal zone and in the cervical T2.

The data collected has been transferred to Matlab, where the translation and rotation of the trunk has been calculated. According to the reference system shown in Fig. 2, the range of motion in each direction has been defined, as shown in Table 1.

Cosimulation of the Patient Behavior and the Rehabilitation Device

The position of continuum manipulators is defined by their kinematics, but also by the external forces applied, as they rely on the deflection of their limbs to reach the position desired. This implies that it is necessary to know the desired position of the jacket and the forces that it must apply on the patient. For that, a model that simulates both the motion of the patient and the jacket has been developed. To validate it, the followed methodology is summarized in Fig. 3. A rigid body kinematic model of the patient has been developed and used to obtain, depending on the degree of disability, the force and moment that the jacket must apply to perform a given motion. Those requirements are fed into the kinematic model of the manipulator to solve the inverse kinematic problem and to obtain angular positions and torque in the motors under quasi static conditions, as it is assumed that the motion will be slow. Thereafter, those motor positions and torques are used to control the motion of the rehabilitation prototype. Finally, after performing several tests, the theoretical and experimentally measured motor torques are compared to check the validity of the models and the performance of the prototype. The axial force in the flexible limbs near the spherical joints are also compared during the validation.

Human Body Kinematic Model and Statics.

The approach proposed in Ref. [26] has been followed to build the human body model. The body is modeled as rigid segments connected by spherical joints with the range of motion restricted and the action of the muscles is represented by torques applied on those joints. To obtain the mass of the body members and the center of mass, solids with a cross section of a disco-rectangle or obround [26] have been used, see Fig. 4. The cross section has been defined by a rectangle with a length of 2t and a width of 2r, and two semicircles located on the lateral of the rectangle with radius r.

These values are obtained from Eq. (1), as a function of the perimeter p and width w of each segment, given by the dimensions of the patient modeled.
t=πwp2π4r=p2w2π4
(1)
After the geometrical values of Eq. (1) are obtained, the mass and centers of mass have been calculated as in Eq. (2), where h the total height of the solid, D the specific density of each segment and A(z) the area of a cross section at a height z measured from the lower section.
M=DA(z)hdzzcm=1MDh2zA(z)dz
(2)

The mathematical model is composed of 17 segments where a series of joints are used to connect them with each other, corresponding to the human joints. Except for the head and hands, where an ellipsoid and spheres have been used, respectively, the segments of the model have been represented with the solid described. The human trunk has been discretized into two segments. A series of constraints have been introduced in the joints of mathematical model where the rotation can be blocked in one two or three axes. According to the planes in which human body is divided, see Fig. 2, constraints are shown in Table 2: abduction and adduction rotations (frontal plane), flexion and extension rotation (sagittal plane) and internal external rotation (transverse plane).

The algorithm of the model has been implemented in Matlab solving the static balance in each segment obtaining the reactions and moments in each joint for a given pose of the body, see Fig. 4. Here, it is assumed that a stroke patient will not provide the forces required to keep the balance in a given position, so the jacket here designed must compensate it. To obtain that compensation, an α factor that represents the degree of deficit of the patient has been proposed, where 0 is full disability and 1 is no disability. Hence, the torque that the manipulator must exert on the trunk to keep the position is obtained as:
τman=(1α)τchest
(3)

Kinematics of the Continuum Parallel Manipulator.

The global reference system XYZ of the manipulator, with origin in O, is located at the same height as the axis of the four motors, see Fig. 5. The couplings between the flexible limbs and the motors axis are in the points Ai (i = 1–4), located at X = e on an ellipse of a x b size. The motor's axis is tangential to the ellipse contour. In a parallel ellipse of the same size, the limbs are joined to the jacket by means of respective spherical joints in Bi, vertically aligned direction with Ai points in the zero position. P is the center of the jacket ellipse. The height between both ellipses OP0 plays a relevant role with respect to the limbs length, more length will be equal to bigger workspace, but can also imply higher deflection and stress on the flexible bars that may be lead to plastic deformation or breakage.

Pseudo-Rigid Body Modeling of the Manipulator Limbs AiBi.

Regarding the modeling of the deflection of the i limbs, given the fact that large deflections are expected, and, taking into account that the limbs will comprise several elements of different geometry and material: R-joint coupling, flexible bars as backbones, bar-sensors coupling, sensors, coupling to S-joints and S-joints, a pseudo-rigid body modeling approach has been taken. The limbs are discretized in N rigid elements joined by spherical joints where a bending moment proportional to the radius of curvature appears.

The position of each element j with respect to the previous one is defined by the azimuthal angle φij and the polar angle θij, as shown in Fig. 6 for N =4 elements. Each limb is fixed by a coupling to the motor axis located in Ai, so, being R the motor axis radius and dL the length of each element, the position Bi of each S-joint with respect to a local reference system XiYiZi located in Ai is shown in Eq. (4), where θi1 is the angular position of the i motor.
AiBiXiYiZi=[010cθi10sθi1sθi10cθi1]{00R+dL}+j=2N[cθijcφijsφijsθijcφijcθijsφijcφijsθijsφijsθij0cθij]{00dL}
(4)
Finally, being [Ri] the rotation matrix that relate the local reference system of each limb with the global reference system, and [R] the rotation matrix from the reference system X′Y′Z′ located in the center of the jacket ellipse, the position P of the center of the jacket is obtained as follows:
OP=[Ri]AiBiXiYiZi+[R]BiPXYZ
(5)
Regarding the calculus of the bending moment, given the fact that the limbs are long, slender and will remain on a plane as there is no torsion due to the S-joint, the Euler–Bernoulli simplification of no shear deformations will be assumed. Hence, the bending moment MKij that appears between two discrete elements j and j + 1 along the length s of the limb will be ruled by Eq. (6).
MKij(s)=E(s)Irad(s)ρ(s)
(6)
In Eq. (6), E is the Young's modulus, I is the second moment of area about the axis perpendicular to the plane, and ρ is the radius of curvature at the section defined by s. The curvature center C is the intersection of two perpendicular lines to the elements j and j + 1 by their centers, so the radius of curvature can be related with the angle βij between the elements as in Eq. (7), see Fig. 7, where dL is the length of each discrete element.
ρ(s)=dL21tanβij2
(7)
Being wij and wij+1 the unit vectors along to the elements, the angle βij between them is calculated in Eq. (8), where the unit vectors are obtained from the third column of the rotation matrix of each element in Eq. (4).
βij=acos(wijwij+1)wij={sθijcφijsθijsφijcθij}T
(8)
Finally, the direction vij of the bending moment will be:
MKij=MKijvij=MKijwij×wij+1|wij×wij+1|
(9)

Inverse Kinematic Problem of the Manipulator.

To solve the inverse kinematic problem, first, using the model developed for a human body, the position of the trunk is set, so the position of the jacket and thus the four OBi S-joint positions are obtained. Solving the static problem, also and applying Eq. (3), the torque that the jacket applies on the patient is calculated. Hence, the four limbs must meet a set of 15 conditions, the three cartesian coordinates of each spherical joint Bi and the three components of the torque τman.

To meet the dynamic condition τman and calculate the deflection of the bars, the static problem of the manipulator must be solved to obtain all the unknown magnitudes. In each bar, there are 5 N +5 unknowns: 2 N – 1 angles θij and φij, as φi1 is 90 deg, see Fig. 6, and 3N joint reactions, 2 components of the moment MA1 in the motor coupling, the motor torque τm1, and the three components of the force in Bi.

Regarding the equations, each element is isolated, and the balance of forces in XYZ and the balance of moments in the two transversal directions to each element is done, as there is no torsion due to the S-joint, with the exception of element j =1. This means that in each bar there are 2 N +1 equations from the balance of moments, 3 N from the balance of forces and 3 from the kinematic condition of the Bi position, hence, a total of 5 N +4 equations per bar. Being 4 flexible bars in the manipulator, 4 additional equations are needed: 3 from the three components of torque condition the manipulator τman must meet. An additional equation is still needed as that the parallel manipulator is redundant. Here, it has been imposed that the vertical component of the resultant force that the bars exert on the jacket nulls, that is, the patient doesn't feel a variable vertical force of the jacket over him, although it doesn't mean that it is the best option to solve the static problem because it might happen that the mechanism must react with other criteria to balance a patient.

Mechatronic Design

Mechanical Design.

After iteration with the kinematic model, a rehabilitation prototype has been developed for preliminary testing. The layout of the device proposed is shown in Fig. 8. An ergonomic jacket, made by 3D printing in ABS+, has been designed based on anthropometric measurements where 95% of the population is able to wear it (P95 percentile). To wear the jacket, two straps has been attached to the back of the design. The only contact surface between jacket and patient is in the back, being possible a system suitable for patients with different sizes. Four supports embedded with spherical joints have been developed to connect the jacket to the flexible mechanism. The dimensions of these pieces are provided according to the kinematic model developed, with dimensions a = 0.52 m, b = 0.25 m, e = 0.28 m, f = 0.21 m, OP0 = 0.39 m and the total length of the limbs from the R-joint to the S-joint is 0.439 m. The S-joints have been made by 3D printing in ABS+, printing ball and socket with a gap lower than 1 mm.

The limbs configuration is shown in Fig. 9. The R-joint coupling, the flexible bars, the coupling with the sensor, the force sensor, the coupling with the S-joint and the S-joint compose each limb of the manipulator. All coupling have been made in Aluminum. The flexible bars have been designed to reach a balance between flexibility and rigidity: the system must have a workspace big enough to achieve the kinematic requirements but must be also capable of applying enough force to the patient. Each group of flexible bars, made of Nylon 6.6 with 6 mm of diameter and a length of 300 mm, have been placed in a circumference of 50 mm of diameter in the section of the coupling. The central bar has a fixed length and the outer bars can slide through the coupling of the motor to keep the parallelism. Four HBM U9C uniaxial sensors have been incorporated to measure tensile and compressive forces.

To place the prototype, a structure has been developed based on a commercial standing frame, commonly used in upper rehabilitation exercises by physiotherapists. The structure has been manufactured using aluminum profiles. The knee and hip supports have also been designed. The knee and hip supports are made by 3D printing in ABS+. Additionally, two side tables have been included where motors have been positioned. Four DC motors Maxon RE 50 with an encoder HEDL 5540 and GP52B mechanical reductor of 81 have been used.

Control System Design.

Figure 10 shows the block diagram of the implemented control system. The control algorithm is a cascaded PID controller where the system input is the trajectory generated from the rotations and displacements obtained in the tests carried out. Four EPOS2 50/5 controllers have been used to control the current and velocity loops. The position loop has been implemented in Labview RT using CompactRio system. The cycle time for the current loop is 0.1 ms, for the velocity loop is 1 ms and for the position loop is 2 ms. In addition, the four motors have been dynamically characterized where the mechanical and friction parameters have been obtained and incorporated them into the PID system. Table 3 shows the parameters of the motors

Experimental Methodology

Several tests have been carried out to check the prototype design and validate the model developed to design it. Three kind of motions were programmed for the trunk: flexion–extension rotation of +1/-10 deg, that is, around the X axis, abduction and adduction rotation of ±8 deg, that is, around Y axis, and internal external rotation of +10/-9 deg, around Z axis in the kinematic model. Two cycles of each motion have been done, with a period of 5 s.

To test the capability of the sensors to monitor the response of the patient, first, the subject was moved by the jacket putting no opposition during the motion. Then, the opposite situation was tested, where the subject tried to counteract the jacket action, to keep it fixed in the initial position. The magnitudes measured were the axial force in the limbs near the S-joint, the angular position of the motors measured by the rotary encoder and the torque of the motors. All the signals have been sampled at 500 Hz and low pass filtered with a cutoff frequency of 20 Hz. To better identify the effect of the subject and the limbs deflection on the measured signals, the same trajectories were also executed removing the jacket and limbs. Without the load, it was possible to isolate the inertial and frictional effects from the motors, and then remove their effect from the torque measurements.

Results and Discussion

First, it is shown a comparison between the tests with the jacket moving and tests counteracting the motion of the jacket, trying to keep it in a fixed position. The measured axial force in S-joints and motor torque in the rotation around X, Y and Z axes are shown, respectively, in Figs. 1113.

Regarding the tests rotating around the X axis, in Fig. 11 left it is seen that the trend of the motors torque is similar when the subject rotates together with the jacket or when he counteracts the jacket motion to keep it fixed. The maximum torque values in motors 1 to 4 with fixed and moving jacket were, respectively, –6.95 and –4.18 Nm, –8.65 Nm and –7.44 Nm, 6.62 Nm and 5.36 Nm, and 5.37 Nm and 4.11 Nm. That means that the counteraction of the subject introduces a variation of 66%, 16%, 23%, and 30% in the maximum torque of the motors.

Comparing the measurements of the force sensors in the limbs in Fig. 11, right, it is seen that when the subject makes no opposition the trends of the four signals is quite similar. However, when the subject counteracts the motion of the jacket, the signals diverge. In the forward limbs, 1 and 2, see numeration in Fig. 5, the traction forces increase to pull from the subject to do the flexion motion, especially at seconds 3.5 and 8.3. At the same time, the limbs in the back, 3 and 4, suffer compression trying to push the subject. This behavior cannot be observed in the torque signals from the motors. At seconds 6 and 11, it happens the opposite trying to do the extension motion. In fact, the force sensors 1 and 2 measure a maximum force in extension with fixed and moving jacket of 37.2 and 11.7 N, 44.8 and 16.2 N, and sensors 3 and 4 measure a maximum in extension of 24.8 and 16.2 N, and 23.2 and 17.8 N, respectively. This means variations of 218%, 176%, 202%, and 383%. Hence, it is clear that the force sensors not only are more sensitive to the patient's response, but also explain better what has happened during the test.

Similar comments can be made for the tests rotating around the Y axis, Fig. 12. The trends in the motor torque signals are similar independently of the reaction of the subject, while the measurements of the force sensors are clearer regarding which limbs are pushing or pulling the patient to do the desired motion. In the abduction motion, Y+, limbs 1 and 4 work under traction to pull the subject while limbs 2 and 3 are compressed pushing him. This is seen at seconds 1.6 and 6.2. In adduction, Y-rotation, the opposite happens, seconds 4 and 9. Comparing the variation of the maximum values of the sensors, the variations are of 32%, 23%, 49%, and 43% in the motors torque, and 177%, 143%, 270%, and 380% in the force measured by the sensors.

Finally, in the tests rotating around Z axis, Fig. 13, the trends of the torque and force signals are similar for both exercises. Figure 13 right indicates that limbs 1 and 3 pull while limbs 2 and 4 push and vice versa to generate the alternating motion. Comparing the variation of the maximum values of the sensors, the variations are of 0.5%, 31%, 5%, and 10% in the motors torque, and 23%, 22%, 7%, and 40% in the force measured by the sensors. As a conclusion, the results prove that, even removing the influence of the motor inertia and friction, the force sensor located in the flexible limbs are more sensitive to the subject performance.

Secondly, the measurements of the tests counteracting the jacket have been compared with the kinematic model. Results of the internal external rotations, around Z axis, are shown in Fig. 14. Comparing the variation of the maximum values of the simulated and experimental signals, the variations are of 19%, 25%, 29%, and 32% in the four motors torque, and 8%, 8%, 32%, and 15% respectively in the force measured by the sensors. These results are explained because of the uncertainty in the test itself, where the subject was trying to counteract the jacket motion, but obviously, there was still some movement. In the model, the jacket simulated was perfectly fixed. Also, Young's modulus of the material of the flexible bars, provided by the manufacturer for a different bar diameter, is another source of uncertainty that has to be checked.

Regarding how the prototype deflects and works on the patient, in Fig. 15 several images have been extracted from the videos taken during the testing. The positions of maximum rotation in each direction are shown. It has been observed that the parallelism of the outer flexible bars is slightly lost under compression, because they slide through the R-joint coupling. Also, some plastic deformations in the bars are seen after several cycles, which is an important aspect to deal with, as the appearance of plastic deformations cannot be allowed in these mechanisms. This also leads to the test of other materials, as, for example, nitinol due to its superelastic properties. Finally, regarding the control and measurement of the patient response, the use of an inclinometer is intended, to have a direct position measurement of the patient and improve the control of his position.

Conclusions

A continuum parallel kinematic prototype for the rehabilitation of trunk motion in patients who have suffered a stroke is proposed here. The prototype proposed is a novel system which aims to study the rehabilitation of the trunk. The design and the model of the prototype are presented in this paper. The kinematic model is based on parallel continuum mechanism where four flexible limbs, which are formed by groups of four flexible bars, are connected to a jacket. The prototype design consists of a wearable jacket connected to four actuated flexible limbs through spherical joints and a standing frame, where the system is placed. The system has been built and several tests have been carried out, based on common rehabilitation exercises of the trunk. Although the results seem to be promising because of the capability of the jacket to move the patient and monitor his performance, there is still room for improvement regarding the modeling approach and parameters, and the sensors to use and their location.

Acknowledgment

Authors want to acknowledge the support given by the Spanish Government (Ministerio de-Economía y Competitividad y Fondo Europeo de-Desarrollo Regional, FEDER, DPI2015-64450-R), and the Basque Government for the project IT949-16. Thanks are addressed to Ion Lascurain-Aguirrebeña and Leire Santisteban, lecturers from the Faculty of Medicine & Infirmary of the University of the Basque Country, Ana Bengoetxea from the Faculté des Sciences dela Motricité from Université Libre deBruxelles, and the staff from the Hospital Gorliz (centre for subacute patients of the Basque public sanitary system Osakidetza) for their valuable advice.

Funding Data

  • Spanish Government: Ministerio deEconomía y Competitividad y Fondo Europeo de-Desarrollo Regional, FEDER (No. DPI2015-64450-R; Funder ID: 10.13039/501100003329).

  • Basque Government: Convocatoria Grupos de-investigación del sistema universitario vasco de tipo A (No. IT949-16).

Nomenclature

     
  • A(z) =

    cross section area in disco-rectangle shape of the human body model parts

  •  
  • Ai =

    position of the R-joints of the parallel manipulator

  •  
  • Bi =

    position of the S-joints of the parallel manipulator

  •  
  • C =

    center of curvature of the radius

  •  
  • D =

    specific density in disco-rectangle shape of the human body model parts

  •  
  • dL =

    length of each discrete element

  •  
  • E =

    Young's modulus of the material of each part of the limbs

  •  
  • h =

    height in disco-rectangle shape of the human body model parts

  •  
  • I =

    index that defines the number of the limb, from 1 to 4

  •  
  • I =

    second moment of area about the axis perpendicular to the plane

  •  
  • j =

    index that defines the number of the discrete element of the limb, from 1 to N

  •  
  • MKij =

    bending moment vector due to the rotation of element j+1 with respect to element j in the limb i

  •  
  • N =

    number of elements used in the discretization of the flexible limbs

  •  
  • O =

    origin of the global reference system XYZ, in the center of the lower ellipse

  •  
  • p =

    perimeter in disco-rectangle shape of the human body model parts

  •  
  • P =

    position of the center of the upper ellipse

  •  
  • r =

    radial dimension in disco-rectangle shape of the human body model parts

  •  
  • R =

    motor axis radius

  •  
  • [R] =

    rotation matrix that relates the local reference system of the jacket with the global reference system

  •  
  • [Ri] =

    rotation matrix that relates the local reference system of each limb with the global reference system

  •  
  • t =

    longitudinal dimension in disco-rectangle shape of the human body model parts

  •  
  • vij =

    unit vector of the bending moment

  •  
  • w =

    width in disco-rectangle shape of the human body model parts

  •  
  • wij =

    longitudinal unit vector of element j from flexible limb i

  •  
  • XiYiZi =

    local reference system of each limb, with origin in Ai

  •  
  • XYZ =

    global reference system

  •  
  • βij =

    angle between elements j and j+1

  •  
  • φij =

    azimuthal angle to locate each discrete element of a flexible limb

  •  
  • θij =

    polar angle to locate each discrete element of a flexible limb. θi1 is also the angular position of the motor axis

  •  
  • ρ(s) =

    radius of curvature at the section defined by s

References

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