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Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 7071064, 20 pages https://doi.org/10.1155/2019/7071064 Research Article Research on an Ankle Joint Auxiliary Rehabilitation Robot with a ′ ′ Rigid-Flexible Hybrid Drive Based on a 2-S PS Mechanism 1 1 1 2 1 Caidong Wang , Liangwen Wang , Tuanhui Wang, Hongpeng Li, Wenliao Du , 1 1 Fannian Meng , and Weiwei Zhang School of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou 450002, China School of Logistics Engineering College, Shanghai Maritime University, Shanghai 200000, China Correspondence should be addressed to Liangwen Wang; firstname.lastname@example.org Received 24 January 2019; Accepted 27 February 2019; Published 17 July 2019 Academic Editor: Alberto Borboni Copyright © 2019 Caidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An ankle joint auxiliary rehabilitation robot has been developed, which consists of an upper platform, a lower platform, a dorsiﬂexion/plantar ﬂexion drive system, a varus/valgus drive system, and some connecting parts. The upper platform connects ′ ′ to the lower platform through a ball pin pair and two driving branch chains based on the S PS mechanism. Although the robot has two degrees of freedom (DOF), the upper platform can realize three kinds of motion. To achieve ankle joint auxiliary rehabilitation, the ankle joint of patients on the upper platform makes a bionic motion. The robot uses a centre ball pin pair as the main support to simulate the motion of the ankle joint; the upper platform and the centre ball pin pair construct a mirror image of a patient’s foot and ankle joint, which satisﬁes the human body physiological characteristics; the driving systems adopt a rigid-ﬂexible hybrid structure; and the dorsiﬂexion/plantar ﬂexion motion and the varus/valgus motion are decoupled. These structural features can avoid secondary damage to the patient. The rehabilitation process is considered, and energy consumption of the robot is studied. An experimental prototype demonstrates that the robot can simulate the motion of the human foot. 1. Introduction pronation-supination and ﬂexion-extension movements. Bi  proposed a spherical parallel kinematic machine as an Many studies have shown that high-intensity repetitive ankle rehabilitation robot, which can improve the adaptabil- movements play an important role in the eﬀectiveness of ity to meet the patient’s needs during rehabilitation. Lu et al.  proposed a three-DOF ankle robot combining passive- robot-assisted therapy . Some ankle rehabilitation robots for treating ankle injuries have been developed. For example, active training. Aggogeri et al.  proposed a new device Roy et al.  developed a three-DOF wearable ankle robot, based on a single-DOF parallel mechanism able to perform back-drivable with low intrinsic mechanical impedance actu- trajectories similar to the patient’s ankle. Erdogan et al.  ated by two actuators. Saglia et al.  designed a redundantly presented a conﬁgurable, powered exoskeleton for ankle actuated parallel mechanism for ankle rehabilitation. Yoon rehabilitation. Liao et al.  proposed a novel hybrid ankle and Ryu  presented a reconﬁgurable ankle rehabilitation rehabilitation robot, which is composed of a serial and a par- robot to cover various rehabilitation exercise modes. Jamwal allel part. The parallel part of the robot was simpliﬁed as a et al.  designed a rehabilitation robot with three-DOF constrained 3-PSP parallel mechanism. The kinematic analy- rotation. The robot has four actuators. Girone et al.  used ses showed that the proposed hybrid rehabilitation robot can a Stewart platform-based system as an ankle robot with six not only realize three kinds of ankle rehabilitation motions DOFs. Veneva  introduced an ankle-foot orthosis with but also eliminate singularity with enhanced workspace. one DOF for the foot segment and another one for the shank Nowadays, research on ankle joint robots involves several segment. Agrawal et al.  designed a two-DOF orthosis with aspects, including control, torque, motion planning, and 2 Applied Bionics and Biomechanics valgus are considered, as they are more important optimization. For example, Rosado et al.  implemented PID controllers in the development of passive rehabilitation for ankle rehabilitation . The basic idea is thus exercises. Meng et al.  presented a robust iterative feed- to develop a rehabilitation robot primarily intended back tuning technique for repetitive training control of a for the above two motions to meet some special compliant parallel ankle rehabilitation robot. Zhang et al. patients’ need and further reduce the cost of produc-  proposed a computational ankle model for use in tion and use robot-assisted therapy estimating the passive ankle torque. (2) The objective is to design a robot in which the motion Ayas et al.  designed a fractional-order controller for a is fully decoupled into motion segments, to avoid the developed 2-DOF parallel ankle rehabilitation robot subject associated motion of multidrive motors to external disturbance to improve the trajectory tracking performance. (3) For existing robots, the motion law of rehabilitation At present, a number of rehabilitation robots are under needs to be further researched. In fact, by applying investigation. However, only very few rehabilitation robots inappropriate rules while rehabilitating the patient, have been commercialised [9, 18, 19]. Rehabilitation robotics the exercise would be less eﬀective and may lead to is penetrating the market very slowly. The signiﬁcant limita- secondary damage to the patient tions are the high cost and the diﬃculty to meet some speciﬁc Based on these considerations, a bionic ankle joint aux- needs from patients. For low-income and middle-income ′ ′ iliary rehabilitation robot based on a 2-S PS mechanism classes, only 5-15% of people who need assistive devices and technologies have access to these technologies . was designed. The main innovation points include There is a shortage of personnel trained to manage the provision of such devices and technologies. However, the (1) The robot is designed with a special structure conﬁg- research and development on rehabilitation robots is emerg- uration. Between the upper platform and the lower ing due to the fact that the cost of excluding people with dis- platform, the centre sphere-pin pair and the two abilities from taking an active part in community life is high drive branch chains used for support are designed and the improvement has to be borne by society, particularly into a right triangle. Among them, the centre for those who take on the burden of care. The following sphere-pin pair is a right-angle vertex, and the two conclusions have been drawn from the literature reviews : drive branch chains are the vertexes of the two right-angle edges of the right triangle. For each drive (1) The existing rehabilitation robots have unacceptably branch chain, the spherical pin shafts of the two high price. Even though limited rehabilitation robots sphere-pin pairs are arranged along the direction of are commercially available, most of them are still the right-angle side of the triangle. The spherical placed at research institutes due to the lack of market pin shaft of the centre sphere-pin pair is also attraction arranged along the direction of another right- (2) Most of the existing ankle rehabilitation robots have angle side of the triangle. Using this innovative coupled motions other than ankle rehabilitation structure conﬁguration, the upper platform realizes needs. On the one hand, it increases the development dorsiﬂexion/plantar ﬂexion and varus/valgus motions cost since unnecessary redundant motions are used. through rotation around two right-angle sides of Most importantly, due to the coupled translations, the triangle additional support will be required to endure the To make the robot motion be completely decoupled patient’s weight. For example, a few ankle rehabilita- during dorsiﬂexion/plantar ﬂexion or varus/valgus tion robots have coupled motions of the legs. It motions, dorsiﬂexion/plantar ﬂexion and varus/ becomes very inconvenient for a patient to sit down valgus driving systems of the robot adopt the rigid- and concentrate on the ankle rehabilitation ﬂexible hybrid structure. The two drive branch chains (3) Most of the existing rehabilitation robots are have the same structure. Each branched chain con- designed for hospital environments sists of a motor, slider block, spring, and others. When Motor I rotates to change Branch Chain 1 There is no indication that patients can operate and tailor which results in motion of the upper platform, the rehabilitation routines to their own needs. A completely Branch Chain 2 will change its length to ﬁt the upper new control mode is in demand which will allow a patient platform motion. The compression of springs on to operate the device by themselves and in the home Branch Chain 2 is large enough to compensate for this environment. kind of change; therefore, Motor II on Branch Chain Therefore, the following three points have been consid- 2 keeps stationary. The same circumstance occurs ered in our designed rehabilitation robot to further improve for Motor II rotating to change Branch Chain 2 their performance and reduce the manufacturing and use (2) The robot uses a centre ball pin pair as the main costs simultaneously: support to simulate the motion of the ankle joint; a (1) Among the three allowed motions of the human structure consisting of the upper platform and the ankle, only dorsiﬂexion/plantar ﬂexion and varus/ centre ball pin pair is a mirror image of a patient’s Applied Bionics and Biomechanics 3 Table 1: Actuation, RoM and motion decoupled characteristic of ankle rehabilitation robots. Year Authors DOF RoM Actuator Motion decouples 41.9 plantarﬂexion 43.8 dorsiﬂexion 42.8 abduction 2006 Liu et al.  3 Electric motor No 41.9 abduction 53.8 inversion 44.1 eversion 50 plantarﬂexion/dorsiﬂexion 2006 Yoon et al.  2 Pneumatic actuator No 55 inversion/eversion 30 dorsiﬂexion 60 plantarﬂexion 2009 Saglia et al.  2 Electric motor No 30 inversion 15 eversion 46 plantarﬂexion/dorsiﬂexion 2009-2014 Jamwal et al. [25, 26] 3 52 abduction/adduction Pneumatic actuator No 26 inversion/eversion 45 plantarﬂexion/dorsiﬂexion 2010 Ding et al.  2 Magneto-rheological ﬂuid (MRF) No 12 inversion/eversion 99.50 inversion/eversion 2013 Bi  3 56.00 dorsiﬂexion/plantarﬂexion Electric motor No 100.80 internal/external rotations 75 plantar/dorsal ﬂexion 2018 Liao et al.  3 Electric motor No 42 inversion/eversion 60 dorsiﬂexion/plantar ﬂexion C.D. Wang 2018 2 Electric motor Yes (author of this paper) 60 varus/valgus foot and ankle joint, which satisﬁes the human body physiological characteristics Compound motion (3) The speed, acceleration, and energy consumption of a typical rehabilitation exercise are considered to select Dorsiflexion/plantar flexion diﬀerent motion laws for the upper platform of the robot, for applying appropriate rules while rehabili- tating the patient and avoiding secondary damage to the patient Table 1 shows actuation, range of motion (RoM), and motion decoupled characteristics about our designed Varus/valgus robot and some stationary ankle rehabilitation robots. From Table 1, only our robot is completely decoupled in dorsiﬂexion/plantar ﬂexion or varus/valgus motions. The rest of the paper is organized as follows: In Section 2, Figure 1: Schematic of the human ankle joint. the structure of the robot is presented. In Section 3, the kine- matic model is established, and the workspace is calculated in Section 4. The motions of the robot’s upper platform are sim- dorsiﬂexion/plantar ﬂexion, varus/valgus, and adduction/ ulated and analyzed under diﬀerent motion laws in Section 5. abduction. Among them, dorsiﬂexion/plantar ﬂexion and The control system and the experimental research are varus/valgus are the two most important . discussed in Section 6. Conclusions are outlined in Section 7. Therefore, an ankle joint auxiliary rehabilitation robot is designed according to the schematic shown in Figure 1. The robot with two drives and two DOFs is capable of three kinds 2. Structure and Working Principle of the Robot of motions, namely, the dorsiﬂexion/plantar ﬂexion motion, According to the anatomical structure of the human ankle, varus/valgus motion, and compound motion. The robot can the ankle involves a total of three kinds of motions, i.e., be used by patients to exercise all these three motions. For 4 Applied Bionics and Biomechanics B (B ) 1 2 B ‵ P′ P n 5 B ‵ l 1 Motor I B ‵ n l A (A ) 1 2 Motor II (a) (b) Figure 2: Schematic of the robot. (a) The robot rotates by an angle of α around the X -axis. (b) Drive branch chain. (1) Lower platform, (2) motor, (3) U-shaped connector, (4) screw rod, (5) guide frame, (6) slider block, (7) spring, (8) upper platform, and (9) ball pin structure. patients with foot droop and lower limb muscle atrophy, the of a patient’s foot and ankle joint. When an ankle joint needs rehabilitation train can recover the ankle activity to normal, to perform the dorsiﬂexion and plantar ﬂexion rehabilitation improve the muscle strength of lower limb muscle atrophy, motions, Motor I starts rotating the screw rod (4) and drives and make the patient stand and walk during the rehabilita- the slider block (6). Then, the motion of the slider block (6) tion period . constricts the spring (7). Under the action of the spring A schematic of the robot is shown in Figure 2, where (1) force, the upper platform rotates by a certain angle along is the lower platform, (8) is the upper platform, and (9) is the the direction of the dorsiﬂexion and plantar ﬂexion reha- ball pin structure supporting the two platforms. The two bilitation motions. drive branch chains (namely, Branch Chain 1, A B and For the dorsiﬂexion/plantar ﬂexion motion, only Motor I 1 1 on branch chain A Branch Chain 2, A B ) are identical; each branched chain B is needed to drive the robot (when 2 2 1 1 consists of a motor (2), U-shaped connector (3), screw rod Motor I rotates to change branch chain A B for making 1 1 (4), guide frame (5), slider block (6), and spring (7). The the dorsiﬂexion/plantar ﬂexion motion, the branch chain screw rod is connected to the motor, which is in turn ﬁxed A B will change its length to ﬁt the upper platform motion. 2 2 on the U-shaped connector. A screw pair is formed by the The compression of springs on branch chain A B is enough 2 2 screw rod and slider block, while the slider block (6) and to compensate for this kind of change). For the varus/valgus guide frame (5) form a sliding pair. Springs are installed rehabilitation motion, the robot operates in the same way between the slider block and guide frame forming the ﬂexible but only Motor II on branch chain A B drives the robot. 2 2 transmission structure. For the compound rehabilitation motion, both motors The guide frame (5) and the upper platform (8) and the drive the device. lower platform (1) and the U-shaped connector (3) are Suppose the patient uses the robot to carry out a reha- connected by ball pin pairs, respectively. Points A , B , A , bilitation motion, the movement time of the dorsiﬂexion/ 1 1 2 B , and O are centre points of the ball pin pairs, and lines plantar ﬂexion, varus/valgus rehabilitation, and compound B B and A A are perpendicular to lines B B and A A , rehabilitation is the same, while the motor power expense 1 3 1 3 3 2 3 2 respectively. It is required that the spherical pin shafts of for the dorsiﬂexion/plantar ﬂexion and varus/valgus rehabil- itation is the same. Compared with the undecoupled robot the ball pin pair A and B lie on the A B B A plane. The 1 1 1 1 3 3 spherical pin shafts of ball pin pair A and B lie on the A motion system, in normal conditions, the robot can reduce 2 2 2 energy consumption of the motor by 30%. B B A plane, and the spherical pin shaft of ball pin O lies 2 3 3 on the A B B A plane. 1 1 3 3 The structural model of the robot is shown in Figure 3. 3. Motion Modelling The foot joint of the patient is buckled on the upper platform (8) using the foot buckle (10). A structure consisting of the The following three cases can be considered when analyzing upper platform and the centric ball pin pair is a mirror image the overall motion of the system: Applied Bionics and Biomechanics 5 Ankle joint Compound motion Dorsiflexion/plantar flexion Varus/valgus − − − B B 1 2 9 6 3 Motor I Motor II A A A 1 3 2 − − − − Figure 3: The structural model of the robot: (1) lower platform, (2) motor, (3) U-shaped connector, (4) screw rod, (5) guide frame, (6) slider block, (7) spring, (8) upper platform, (9) ball pin structure, and (10) foot buckle. Case 1. Only Motor I rotates. X Y Z is set up with point O as the origin, and the coordi- 1 1 1 nate system is ﬁxed to the upper platform. The X -axis is parallel to B B , the Y -axis is parallel to B B , and the 2 3 1 1 3 Case 2. Only Motor II rotates. Z -axis coincides with OA . The coordinates of the points 1 3 1 1 on the upper platform are B 0, n, l , B b,0, l , and 1 3 2 3 Case 3. Motor I and Motor II rotate simultaneously. B 0, 0, l . Due to the symmetrical structure of the upper 3 3 platform, it is considered approximately that the mass centre Each of the above cases can be divided into three modes: point of the upper platform is point P in the middle of line B B and its coordinates are P 0, n/2, l . In the initial 1 3 3 (1) A transition mode (i.e., the motor rotates and state, the two coordinate systems X Y Z and X Y Z 0 0 0 1 1 1 compresses the spring but cannot drive the motion are coincident. of the upper platform) According to Figure 2, when the upper platform rotates by an angle α around the shaft X , the homoge- (2) A rigid-ﬂexible combination driving mode (i.e., the neous transformation matrix is given by motor rotates and compresses the spring, which drives the upper platform) 10 0 0 (3) A rigid driving mode (i.e., the motor rotates and the spring is compressed to a rigid body, which drives 0 cos α −sin α 0 T = 1 the upper platform) 0 sin α cos α 0 The transition mode is not considered, and the rigid 00 0 1 driving mode cannot occur in a normal working state. Case 1 is used as an example to analyze the motion of the robot. 0 0 0 0 1 0 1 0 1 From Figure 2, B , B , and B are the initial positions ′ ′ ′ We have B = T ⋅ B , B = T ⋅ B , B = T ⋅ B , 1 2 3 1 1 2 2 3 3 1 1 1 ′ ′ ′ 0 of the upper platform, while B , B , and B are the corre- 1 2 3 and P = T ⋅ P. sponding ﬁnal positions. In order to calculate relationship between the motor In the initial state, let A B = l , A B = l , OB = 1 1 1 2 2 2 3 drive angle and the motion angle of the upper platform, the l , A A = B B = n, and A A = B B = b. Set a ﬁxed 3 1 3 1 3 2 3 2 3 calculation steps are as follows. coordinate system X Y Z with the centre point O of the 0 0 0 centre ball pin as the origin. The coordinate system is ﬁxed (1) Calculating Initial Compression Displacements of the to the lower platform. X -axis is parallel to A A , Y -axis is Four Springs. Displacements x and x and x 0 2 3 0 12 22 32 parallel to A A , and Z -axis coincides with OA . The coordi- and x represent the initial compression displace- 1 3 0 3 nates of the points on the lower platform are A 0, n,−l , ments of the upper spring and the lower spring of 1 4 0 0 A b,0,−l , and A 0, 0,−l . A motion coordinate system Branch Chain 1 and Branch Chain 2, respectively. 2 4 3 4 6 Applied Bionics and Biomechanics According to the forces and loads on the upper the upper platform rotates by an angle α around platform, establish equilibrium equations for forces the shaft X with the centre point O, and the and torques and the initial compression displace- rotation angle of Motor I is about φ . ments x , x , x , and x of the four springs can 12 22 32 42 be calculated The acceleration P′ of the mass centre point P (see Figure 2) for the upper platform is as (2) Calculating Acceleration of the Mass Centre follows: PointP. When Motor I rotates for a time t s , X o / =0, n cos α ⋅ α + sin α ⋅ α Y / = + l sin α ⋅ α + cos α ⋅ α , P 3 2 n sin α ⋅ α + cos α ⋅ α Z / = l cos α ⋅ α + sin α ⋅ α , P 3 where α, α, and α are the angle displacement, angle respectively; and k is the elastic coeﬃcient of the velocity, and angle acceleration of the upper platform spring. The elastic coeﬃcients for the upper spring rotation motion, respectively and the lower spring are assumed to be the same. (3) Calculating Acting Force between Branch Chain 1 and Establish the diﬀerential equation for guide frame (5). the Upper Platform. The force between Branch Chain According to the initial conditions, t =0, Δx =0, and 1 and the upper platform is F , rotational inertia of Δx =0, we have B13 2 the upper platform around the X -axis is J , and 0 X E cos 2K/m t E weight of the upper platform is m. Considering the 2 1 Δx = − + , 4 2K 2K inertia force and inertia moment of each part of Branch Chain 1, the equilibrium equation is estab- where E =2K ⋅ Δx + K x − x − F − A ⋅ G . 2 1 12 22 B13 12 1 lished and F is solved B13 F represents the reaction force of the upper B13 (4) Calculating Acting Force of the Upper Spring and platform on the guide frame, F = −F ; m B13 1 B13 the Lower Spring of Branch Chain 1. In the A B is the mass of the slider block; G is the gravity direction, suppose the slider block rises Δx and of the slider block; A = n sin α + l cos α + l / the guide frame rises Δx . Then, the compression 12 3 4 value (see Figure 2) for the spring is Δx − Δx . 2 2 1 2 n cos α − l sin α − n + n sin α + l cos α + l ; 3 3 4 Therefore, we have n, l , and l are the structural parameters of the 3 4 robot; and α is the angle displacement of the upper F = F =Kx + Δx − Δx , 13 13 12 1 2 platform rotation motion F = F =Kx − Δx − Δx , 23 23 22 1 2 (5) Calculating Angle Relation between the Upper Plat- form Motion and the Motor Drive. When Motor I where F and F are the forces of the upper rotates by an angle φ for a time t s , and the moving spring of Branch Chain 1 on the guide frame and displacement of the slider block is Δx = φ s /2π, 1 n on the guide block, respectively; F and F are 23 23 then s is the screw pitch of the screw rod (4). the forces of the lower spring of Branch Chain 1 on the guide frame and on the slider block, Thus, we can obtain D − l − x − x /2 − F /2K − G ⋅ A /2K 1 − cos 2K/m t 12 1 12 22 B13 1 12 1 φ = , 5 s 1 − cos 2K/m t /2π n 1 Applied Bionics and Biomechanics 7 2 2 where t ≠ 0, D = n cos α − l sin α − n + n sin α + l cos α + l , 12 3 3 4 and F = −F . Other parameters are the same as B13 B13 those for equations (2), (3), and (4) 4. Solving the Workspace We use the movement locus of the centre point P on the upper platform to express workspaces of the upper platform. For solving the workspace, the numerical method and ana- lytical method are combined. Taking Motor I as an example, ° 70 with the upper platform rotating by an angle α , calculate the lengths l α and l α of Branch Chains 1 and 2, respec- 1 2 tively, at a given angle and evaluate whether or not l α and l α are between the shortest and longest ranges of 2 l allowed branch chains. If they are within an attainable range, Branch Chains 1 and 2 with lengths l α and l α , respec- 1 2 tively, may form a position of the upper platform. By contin- uously changing the angle α and evaluating the results, diverse positions for the upper platform can be obtained, corresponding to the workspace of the upper platform when Motor I runs. Similarly, the workspace for Motor II can be 10 obtained. For solving the workspace when Motors I and II work jointly, ﬁrst, the working space for each motor running solely must be obtained. Then, the two working spaces are aggregated. 4.1. Structure Constraints of the Branch Chain. The structural model of a branch chain is shown in Figure 4. Figure 4: Structural model of the individual branch chain. Its overall length is l (hereafter referred to as the rod length), solid length of the upper spring is l , solid length of the lower spring is l , length of the guide frame is l , 2 60 30 l + l − ε − l − l + l + l 10 40 l40 80 50 30 20 and distance between the top of the guide frame and the = n cos α − l sin α − n upper platform is l . Distance between the U-shaped max 3 max connector and the lower platform is l , length of the screw + n sin α + l cos α + l max 3 max 4 rod is l , and width of the slider block is l . At the initial 40 80 position, the length between the centre of the slider block (2) When Branch Chain II determines the motion, the and the lower edge of the guide frame is l , distance between maximum and minimum angles meet the following the lower edge of the guide frame and the upper edge of conditions: the U-shaped connector is l , and l is the thickness of 90 6 the guide frame. l + l + l + l 10 40 20 6 2 2 4.2. Limit Angles of the Upper Platform for Solving the = l sin α + l cos α + l , 3 min 3 min 4 Workspace. In our research, a workspace computational model is established using the workspace of the centre point P on the upper platform as the robot’s workspace; the dorsi- ﬂexion/plantar ﬂexion motion is taken as an example to l + l + l − − l − l + l + l explain limit angles of the upper platform for solving the 10 90 70 50 6 30 20 workspace. 2 2 = l sin α + l cos α + l 3 max 3 max 4 (1) When Branch Chain I determines the motion, the maximum and minimum angles meet the following conditions: The minimum rotation angles are calculated by equa- tions (6) and (8), and the maximum rotation angles are cal- 2 2 culated by equations (7) and (9). For the calculation results, l + l + l + l = n cos α − l sin α − n 10 40 20 6 min 3 min 2 the absolute value of the minimum or the maximum angle + n sin α + l cos α + l , min 3 min 4 is a limit angle of the upper platform for the dorsiﬂexion/ plantar ﬂexion motion. Y (mm) 8 Applied Bionics and Biomechanics 3 5 47.11° b z′ O′ y′ 7 x′ 30° l P 3 8 Figure 5: A posture model of ankle joint rehabilitation motion: (1) Figure 6: Limit posture of the ankle joint for the dorsiﬂexion body, (2) hip joint, (3) thigh, (4) knee joint, (5) shank, (6) ankle motion (α =30 ). max joint, (7) foot, (8) upper platform, and (9) ball pin structure. 4.3. Calculation Examples and Discussion. Based on the above analysis, a solving system for the workspace is established. Let n = 150, b = 150, l = 282, l =33, l = 295, l = 289, 10 20 30 40 46.71° ε =2, l =63, l =24, l =10, l =73, l =15, and l = l40 50 60 6 70 80 90 94. In the dorsiﬂexion/plantar ﬂexion motion, α =30 max ° ° and α = −30 , and in the varus/valgus motion, β =30 min max and β = −30 . We study the changes of ankle posture in min rehabilitation motion. Take adult males in China as an example: according to −30° the National Report on Nutrition and Chronic Diseases of Chinese Residents and New National Standard of Human Dimensions of Chinese Adults, the adult male has a thigh Figure 7: Limit posture of the ankle joint for the plantar ﬂexion length of 465 mm, a shank length of 369 mm, and a medial motion (α = −30 ). max malleolus height of 112 mm [29, 30]. Based on the human dimensions of Chinese adults, establish a posture model of ankle joint rehabilitation motion as shown in Figure 5 and make simulation analysis in SOLIDWORKS. In Figure 5, the dimensions are a thigh length of l = 465mm, a shank length of l = 369 mm, a medial malleolus height of l = 127 mm, and ball pin structure height l =45 mm. c 3 When an ankle joint carries out the dorsiﬂexion/plantar ﬂexion motion, the upper platform is driven by Motor I ° ° and rotates around the X-axis at α =30 and α = −30 ; max min limit postures of the ankle joint for the dorsiﬂexion motion −20 and the plantar ﬂexion motion are as shown in Figures 6 and 7. −40 Here, we use the movement locus of mass centre point P 80 on the upper platform to express the workspaces of the upper 1 0.5 platform. The workspaces are shown in Figure 8 for the dorsiﬂexion/plantar ﬂexion motion. −0.5 50 −1 When an ankle joint carries out the varus/valgus motion, the upper platform is driven by Motor II and rotates around Figure 8: Working space for the dorsiﬂexion/plantar ﬂexion ° ° the Y-axis at β =30 and β = −30 ; limit postures of the max min motion. ankle joint for the varus motion and valgus motion are as shown in Figures 9 and 10. The workspaces are shown in 5. Motion Simulation for the Upper Platform Figure 11 for the varus/valgus motion. When an ankle joint carries out the compound motion, Driven by Different Motion Laws the robot is driven by the associated motion of Motor I and Motor II. The workspaces of the upper platform are shown The performance of the robot is studied using the follow- in Figure 12. ing three motion laws of the upper platform: modiﬁed X (mm) Z (mm) Y (mm) Applied Bionics and Biomechanics 9 36.76° 12.5 11.5 75.5 30° 75 74.5 −5 −10 Figure 11: Working space for the varus/valgus motion. dS t Figure 9: Limit posture of the ankle joint for the varus motion V = = v, dT h (β =30 ). max d S t A = = a, 10 dT where h and t are the total displacement and total time of the motion phase, respectively; time t varies in 0, t , and when t = t , s = h.Ranges of T and S are 0, 1 . Figure 13 shows a general harmonic trapezoidal motion 36.88° law expressed in dimensionless quantities. The curve is composed of seven sections, and the acceler- ation of each segment is expressed as T π A sin ⋅ 0 ≤ T ≤ T , 1 1 T 2 A T < T ≤ T , 1 1 2 π T − T A cos T < T ≤ T , 1 2 3 P 2 T − T 3 2 −30° A = 0 T < T ≤ T , 11 3 4 π T − T −A sin T < T ≤ T , 2 4 5 2 T − T 5 4 −A T < T ≤ T , 2 5 6 Figure 10: Limit posture of the ankle joint for the valgus motion (β = −30 ). π T − T min −A cos T < T ≤ T , 2 6 7 2 T − T 7 6 trapezoid, modiﬁed constant velocity, and modiﬁed sine motion law . Motion parameters are treated by dimensionless process- ing. The terms t, s, v,and a are the time, displacement, veloc- (1) By choosing diﬀerent T , the three motion laws listed ity, and acceleration, respectively, of the motion laws. The in Table 2 can be obtained. For T = T , according to terms T, S, V,and A are the corresponding dimensionless equation (11), we have parameters, and the following relationship can be established: A = A =AT 12 i i T = , By integrating equation (12) twice, and substituting the boundary condition, i.e., T =0, S =0, and V =0 and T =1, S =1, and V =0, and the continuous vari- S = , h ation conditions of the motion variables in motion X (mm) Z (mm) Y 10 Applied Bionics and Biomechanics −20 −40 −20 50 −40 Figure 12: Working space for Motor I and Motor II working simultaneously. a relationship between the rotating angle φ of the motor and the time t can be calculated. Elastic coeﬃcients of the upper spring and the lower spring might diﬀer in the driven branch chain. To simplify T T T T T T T T T 0 1 2 3 4 5 6 7 the problem, when calculating the driving function of the motor, the elastic coeﬃcients of the upper spring and the lower spring are selected with an identical value. ADAMS software was used to simulate the motion of the upper platform. The parameters are as follows: n = T T T T 4 5 6 7 150 mm, b = 150 mm, l = 704 1 mm, l =12 7 mm, and l = 1 3 4 T T T T T 0 1 2 3 691 4 mm; load on the upper platform is 2 kg; rotational iner- −A tia circling around X-axis is J =43 175 kg·mm ; elasticity coeﬃcient of the upper springs is K =5 5125 N/mm; elastic- Figure 13: A general harmonic trapezoidal curve. ity coeﬃcient of the lower springs is K =7 4059 N/mm; screw pitch of the screw rod is s =5 mm; and weights of the process, we have upper platform and guide frame are m =6 537 kg and m =1 269 kg, respectively. Here, only the simulation analy- S = S =ST 13 i i sis of the dorsiﬂexion/plantar ﬂexion motion is given. The three-dimensional model of the robot is imported into ADAMS software (Figure 14). Revolving joint motion (2) In order to calculate a motor drive function, the around the Z-axis is added to the motor to simulate the maximum motion angle of the upper platform is motor’s rotation. α and time is t . At time t 0 ≤ t ≤ t , according max h i i h When only Motor І rotates, the upper platform is to T = t /t , we have T i i h i loading and the simulation is given here. In the work pro- According to the motion law chosen for the upper plat- cess, the upper platform adopts the modiﬁed trapezoid, form, S is calculated by equation (13) and the motion angle the modiﬁed constant velocity, and the modiﬁed sinusoidal α of the upper platform is calculated as follows: motion laws. A cuboid whose outline size is 220 × 60 × 40 mm α = S α 14 (L × H × W) is added to the upper platform, and a 2 kg mass i i max is set to simulate the patient’s foot. The simulation time of the ° ° upward motion (i.e., α changes from 0 to 30 ) or downward Then, equation (14) is substituted into equation (5), and X (mm) Z (mm) Applied Bionics and Biomechanics 11 Table 2: Diﬀerent motion laws. T T T T T T T T 0 1 2 3 4 5 6 7 Modiﬁed trapezoid motion law 0 1/8 3/8 1/2 1/2 5/8 7/8 1 Modiﬁed sinusoidal motion law 0 1/8 1/8 1/2 1/2 7/8 7/8 1 Modiﬁed constant velocity motion law 0 1/16 1/16 1/4 3/4 15/16 15/16 1 Figure 14: Importing the model into ADAMS. ° ° motion (α changes from 0 to -30 ) of the upper platform is When the spring is set to the elastic state and the rigid 5 s in steps of 0.1 s. state, simulation analysis is carried out. Some of the simu- The upper platform is driven by motion laws previously lated parameters are summarized in Table 3. For the spring in the elastic state, the maximum angular established, and the motor torque changes are shown in Figures 15 and 16 when the upper platform moves from the velocity is 0.234 rad/s and the maximum angular acceleration equilibrium position upward to the top position and down- is -1.290 rad/s , whereas for the spring in the rigid state, the ward to the lowest position, respectively. maximum angular velocity is 0.229 rad/s and the maximum When the motion is driven by the modiﬁed trapezoidal angular acceleration is -0.593 rad/s . The maximum angular function or the modiﬁed constant velocity function, the velocity value of the upper platform moving with the same torque values of the motor ﬂuctuate at the beginning, motion law is larger for the spring in the case of the elastic middle, and end of the motion. The use of the modiﬁed state than that for the spring in the case of the rigid state. sine function enjoys better results than the other two Moreover, the maximum angular acceleration value of the kinds of driving function. upper platform is signiﬁcantly higher for the spring in the 12 Applied Bionics and Biomechanics 25.0 20.0 15.0 10.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 t (s) MOTION_1.Element_Torque.Mag MOTION_2.Element_Torque.Mag MOTION_3.Element_Torque.Mag Figure 15: Torque for the upper platform moving upward. MOTION_1: modiﬁed constant velocity function; MOTION_2: modiﬁed sine function, MOTION_3: modiﬁed trapezoidal function. 35.0 30.0 25.0 20.0 15.0 0.0 1.0 2.0 3.0 4.0 5.0 t (s) MOTION_1.Element_Torque.Mag MOTION_2.Element_Torque.Mag MOTION_3.Element_Torque.Mag Figure 16: Torque for the upper platform moving downward. MOTION_5: modiﬁed constant velocity function; MOTION_6: modiﬁed sine function; MOTION_7: modiﬁed trapezoidal function. Table 3: Analysis results of maximum angular speed and acceleration of the platform and maximum motor torque. Project Maximum angular Maximum angular Motor maximum Curve velocity (rad/s) acceleration (rad/s ) torque (N·mm) Upward Downward Upward Downward Upward Downward motion motion motion motion motion motion Modiﬁed trapezoid 0.234 -0.223 -0.756 0.503 0.382 0.534 Spring in the elastic state Modiﬁed constant velocity 0.164 -0.155 -0.222 -1.290 0.401 0.560 Modiﬁed sinusoidal 0.208 -0.197 0.200 0.440 0.384 0.538 Modiﬁed trapezoid 0.229 -0.222 0.414 -0.538 0.387 0.539 Spring in the rigid state Modiﬁed constant velocity 0.157 -0.150 -0.209 -0.555 0.407 0.559 Modiﬁed sinusoidal 0.201 -0.194 -0.593 -0.197 0.391 0.541 Newton-mm Newton-mm Applied Bionics and Biomechanics 13 case of the elastic state. However, the maximum value of the PC torque does not signiﬁcantly diﬀer for the elastic state and the rigid state. Therefore, in the rehabilitation exercise, the Multiaxis motion patient can choose the diﬀerent laws of motion based on Photoelectric Photoelectric switch 1 control card switch 2 the speciﬁc rehabilitation needs. Servo Servo 6. Control System and Prototype Experiment ServomotorI Servomotor II driver driver 1 2 6.1. Control System Scheme. The control system shown in Figure 17 is composed of a PC, a multiaxis motion control Figure 17: Schematic of the overall control system. card, and servo drive control systems. The PC provides the user with a graphical interface to complete diﬀerent tasks The maximum working angles for the varus/valgus such as the motion parameter setting. The multiaxis motion rehabilitation motion according to the modiﬁed sine motion control card obtains the instructions and then converts them law are shown in Figure 21. into the corresponding signals. The servo driver receives the corresponding signals and drives the servomotor. 6.2.4. The Real-Time Process of Rehabilitation Motion. The varus/valgus motion, the dorsiﬂexion/plantar ﬂexion motion, 6.2. Prototype Test. We design a pose measurement sys- and the compound motion are tested. Here, only the varus/ tem. The measurement system uses a gyro accelerometer valgus motions are used as an example. The rehabilitation MPU6050 to measure the motion angle of the upper platform motion of the upper platform is driven by the modiﬁed sine and a power analyzer HIOKI PW6001 to measure the motion law, and the cycle time is 20 s. The experiment results currents and power of the motor. Measurement data is ° ° ° for three working angles (changing from -10 to +10 , -15 to shown through the PC. The measurement system can display ° ° ° +15 , and -20 to +20 ) are shown in Figures 22, 23, and 24. the upper platform movement in three-dimensional angle The theoretical values in Figures 22, 23, and 24 are changes. Table 4 summarizes the main technical parameters simulated using ADAMS software. From the test results, we of the servomotor used. The robot experimental prototype found that the overall trends of the actual value were con- and the measurement system are shown in Figure 18. sistent with the simulation results. To compare experiment results with the simulation The actual working angles deviate from the ideal value results using ADAMS software, a cuboid load with overall ° ° between -1.7 and +1.6 , when the working angles change dimensions of 190 × 130 × 50 mm (L × H × W) and weight ° ° from -10 to +10 as shown in Figure 22. The actual working of 2 kg is added on the platform to simulate the patient foot. ° ° angles deviate from the ideal value by -1.2 to+1.0 , when the ° ° working angles change from -15 to +15 as shown in 6.2.1. Single Motor Drives. Figure 19 shows the angle changes Figure 23. The actual working angles deviate from the ideal of the upper platform when the robot is driven by Motor I ° ° value by -1.1 to+0.6 , when the working angles change from (see Figure 3) to realize the dorsiﬂexion/plantar ﬂexion ° ° -20 to +20 as shown in Figure 24. movement. Table 5 shows the angle changes of the upper The experiment result for the three working velocities platform. corresponding to the angles changing in Figures 22, 23, and From Figure 19 and Table 5, the upper platform only 24 is shown in Figure 25. The characteristic value of the conducts angle changes needed for the dorsiﬂexion/plantar working velocities is shown in Table 6. The actual velocity ﬂexion movement. Experiments for the varus/valgus move- values are obtained by diﬀerential calculation from the actual ment have the same result. Those experiments show that working angle change values. While from Figure 25 the the experimental prototype of the robot can realize drive varus/valgus rehabilitation motion is not smooth, there are motion decoupling. some velocity ﬂuctuations. From Table 6, theoretical values of the velocity are obtained by calculating from the modiﬁed 6.2.2. The Compound Motion. Realizing the compound sine motion law used by the upper platform motion; the motion is tested by using two motor drives. Figure 20 actual testing maximum value and minimum value of the shows the angle changes of the upper platform when the velocity are larger than the ones of the theoretical velocity. robot is driven using two motors. From Figure 20, the The test results show that the speed ﬂuctuates greatly when upper platform can conduct the angle changes needed the upper platform moves to the extreme position and hori- for the compound motion. zontal position. This result is caused by the rigid-ﬂexible hybrid structure of the robot. The spring is subjected to the 6.2.3. The Maximum Working Angles for the Upper Platform. pressing force which causes it to ﬂuctuate in the abovemen- According to the design parameters, the maximum working tioned stage, causing deformation ﬂuctuations. angles for the dorsiﬂexion/plantar ﬂexion motion or the A power analyzer Hioki PW6001 is used to measure the ° ° varus/valgus motion change from -30 to +30 . Actual maxi- working currents of the motor. A working interface of the mum working angles for the upper platform are tested. The power analyzer is shown in Figure 26. The working currents experiment shows that the maximum working angles meet of the motor for the varus/valgus rehabilitation motion (from ° ° the design requirements. -20 to +20 ) are shown in Figure 27. The maximum value of 14 Applied Bionics and Biomechanics Table 4: Technical parameters of the servomotor. Category Parameter Category Parameter Motor model ACH-06040DC Maximum torque 3.8 N·m Rated power 400 W Rated line current 2.8 A Rated speed 3000 r/min Rated line voltage 220 V Rated torque 1.27 N·m Number of encoder lines 2500 PPR 1234567 Table 5: Angle changes for the upper platform. ° ° ° α ( ) β ( ) γ ( ) Number Around the Around the Around the shaft X shaft Y shaft Z 0 0 0 1 -2.5763 0.0165 -0.0055 2 -3.3618 0.0275 -0.0055 3 -4.1473 0.0385 -0.0055 4 -5.0098 0.0439 -0.0055 5 -5.8667 0.0494 -0.0110 6 -6.8005 0.0439 -0.0055 7 -7.7069 0.0439 -0.0055 8 -8.5034 0.0604 -0.0110 9 -9.2889 0.0769 -0.0110 10 -9.9207 0.0989 -0.0165 11 -10.5414 0.1099 -0.0165 12 -11.0083 0.1263 -0.0220 Figure 18: Experimental prototype of the robot and measurement 13 -11.4203 0.1373 -0.0220 system: (1) control cabinet, (2) gyro, (3) prototype of the robot, (4) load simulating patient foot, (5) PC for the control system of the robot, (6) PC for the measurement system, and (7) power analyzer. Figure 20: The experiment for the compound motion. the currents is 2.62 A. The working currents of the motor for ° ° the varus/valgus rehabilitation motion (from -10 to +10 ) are shown in Figure 28, and the maximum value of the currents is 2.21 A. From Table 4, the rated line current of the servomo- tor is 2.8 A, which indicates that the motor works in the normal range. The current changes periodically, and its period is basically the same with the speed period. The test Figure 19: The experiment for a single motor drive. Applied Bionics and Biomechanics 15 30.36º −30.45º (a) (b) Figure 21: The maximum working angles for the varus/valgus rehabilitation motion: (a) move up and (b) move down. 10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 −2 t (s) −4 −6 −8 −10 −12 Actual value Theoretical value ° ° Figure 22: The working angles (from -10 to +10 ) for the varus/valgus rehabilitation motion. 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 −2 t (s) −4 −6 −8 −10 −12 −14 −16 Actual value Theoretical value ° ° Figure 23: The working angles (from -15 to +15 ) for the varus/valgus rehabilitation motion. Degree (º) Degree (º) 16 Applied Bionics and Biomechanics −2 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 −4 t (s) −6 −8 −10 −12 −14 −16 −18 −20 Actual value Theoretical value ° ° Figure 24: The working angles (from -20 to +20 ) for the varus/valgus rehabilitation motion. 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 −1 t (s) −2 −3 −4 −5 −6 −7 −8 20º 15º 10º Figure 25: The working velocities for the varus/valgus rehabilitation motion. Table 6: The angular velocity characteristic value changes for the upper platform. ° ° ° ° ° ° ° Working angles ( ) -20 to +20 -15 to +15 -10 to +10 Angular velocity ( /s) Theoretical value Actual value Theoretical value Actual value Theoretical value Actual value Maximum value 6.978 7.624 5.233 5.658 3.489 4.175 Minimum value -6.978 -7.503 -5.233 -5.966 -3.489 -3.850 results show that the current is relatively stable at 8.6-10 s. may be caused by manufacturing and assembling precision This is due to the fact that the upper platform moves close for the structure, especially the manufacturing precision of to the horizontal position and the upper platform moves at the spring, the screw, etc. The performance of the spring is a lower speed. The load of the platform is mainly carried by a critical factor. the ball pin pair, and the load component of the varus/valgus branch chain is small and the change is not obvious. 6.2.5. Rehabilitation Motion on a Human Ankle Joint. We According to the on-the-spot observation and test, the tested the robot on a human ankle joint in the lab; the test error between the actual values and the theoretical values scenario is as shown in Figure 29. Degree/t(º/s) Degree (º) Applied Bionics and Biomechanics 17 Figure 26: A working interface of the power analyzer. 2.5 1.5 0.5 −0.5 −1 −1.5 −2 −2.5 −3 0123456789 10 11 12 13 14 15 16 17 18 19 20 t (s) Figure 27: The working currents of the motor for the varus/valgus ° ° rehabilitation motion (from -20 to +20 ). Figure 29: Testing the robot on a human ankle joint. 2.5 We have tested the varus/valgus motion, the dorsiﬂexion/ 1.5 plantar ﬂexion motion, and the compound motion, sepa- rately. The rehabilitation motion of the upper platform is driven by the modiﬁed sine motion law, and the cycle time is 20 s. 0.5 Here, only the varus/valgus motion (working angles ° ° changing from -15 to +15 ) is used as an example. The experiment results are shown in Figure 30. The actual value ° ° −0.5 deviates from the ideal value by -2.1 to +0.9 . Analyzing the result in Figure 30, we found that the −1 overall trends of the actual results tested on the human ankle joint are consistent with the theoretical values. Load on the −1.5 upper platform for the human ankle joint is 7.2 kg. Compar- ing this result with the result tested on adopting the cuboid −2 0123456789 10 11 12 13 14 15 16 17 18 19 20 load (cuboid load is 2 kg, as shown in Figure 18), there are t (s) small diﬀerences. A further in-depth study about clinical data is our future Figure 28: The working currents of the motor for the Varus/valgus ° ° work target. rehabilitation motion (from -10 to +10 ). Current (A) Current (A) 18 Applied Bionics and Biomechanics −1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 −3 t (s) −5 −7 −9 −11 −13 −15 Actual test value eoretical value ° ° Figure 30: The working angles (from -15 to +15 ) for the varus/valgus rehabilitation motion tested on a human ankle joint. 7. Conclusion Δx : The rising displacement of the guide frame F : The force of the upper spring of Branch Chain 1 on the This paper presents an ankle joint rehabilitation robot guide frame with a rigid-ﬂexible hybrid driving structure based on a ′ The force of the upper spring of Branch Chain 1 on the F : ′ ′ 2-S PS mechanism. The robot has two DOFs but can guide block realize the three kinds of motion for the ankle joint F : The force of the lower spring of Branch Chain 1 on the rehabilitation. guide frame The robot uses a centre ball pin pair as the main support The forces of the lower spring of Branch Chain 1 on F : to reduce the load of the drive system. The structure of the slider block the robot consisting of an upper platform and a centre K: The elastic coeﬃcient of the spring ball pin pair is a mirror image of a patient’s foot and m : The mass of the slider block ankle joint, which accords with physiological characteris- G : The gravity of the slider block tics of the human body. In the dorsiﬂexion/plantar ﬂex- n: The structural size of the upper platform ion or varus/valgus driving system, the robot adopts the b: The structural size of the upper platform rigid-ﬂexible hybrid structure and the robot motion is l : The structure height of the ball pin completely decoupled. l : The length between the centre point O of the centre The presented robot has low manufacturing and usage ball pin and the lower platform costs. The theoretical analysis and experimental prototype s : The screw pitch of the screw rod show that the robot can meet some rehabilitation needs of l : The overall length of the branch chain diﬀerent patients. l : The overall length of the branch chain in the initial state l : The solid length of the upper spring Nomenclature l : The solid length of the lower spring l : The length of the guide frame Acceleration of the mass centre point P P′: 30 l : The distance between the top of the guide frame and x : The initial compression displacements of the upper the upper platform spring of Branch Chain 1 l : The distance between the U-shaped connector and the x : The initial compression displacements of the lower 10 lower platform spring of Branch Chain 1 l : The length of the screw rod x : The initial compression displacements of the upper 40 l : The width of the slider block spring of Branch Chain 2 l : The length between the centre of the slider block and x : The initial compression displacements of the lower 70 the lower edge of the guide frame at the initial position spring of Branch Chain 2 l : The distance between the lower edge of the guide F : The force between Branch Chain 1 and the upper B13 frame and the upper edge of the U-shaped connector platform l : The thickness of the guide frame ′ The reaction force of the upper platform on the guide F : B13 l : The length of a thigh frame a l : The length of a shank J : The rotational inertia of the upper platform around b l : The height of the medial malleolus the X -axis c h: The total displacement of the motion phase m: The weight of the upper platform t : The total time of the motion phase. Δx : The rising displacement of the slider block h Deg(º) Applied Bionics and Biomechanics 19 Data Availability Applied Bionics and Biomechanics, vol. 2016, Article ID 9707801, 8 pages, 2016. The data used to support the ﬁndings of this study are  A. Erdogan, B. Celebi, A. C. Satici, and V. Patoglu, “Assist available from the corresponding author upon request. on-ankle: a reconﬁgurable ankle exoskeleton with series- elastic actuation,” Autonomous Robots, vol. 41, no. 3, pp. 743–758, 2017. Conflicts of Interest  Z. Liao, L. Yao, Z. Lu, and J. Zhang, “Screw theory based math- ematical modeling and kinematic analysis of a novel ankle The authors declare that they have no conﬂicts of interest. rehabilitation robot with a constrained 3-PSP mechanism topology,” International Journal of Intelligent Robotics and Acknowledgments Applications, vol. 2, no. 3, pp. 351–360, 2018.  W. M. A. Rosado, L. G. V. Valdes, A. B. Ortega, J. R. 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Applied Bionics and Biomechanics – Hindawi Publishing Corporation
Published: Jul 17, 2019
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