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Abstract
applied sciences Article A 4-DOF Workspace Lower Limb Rehabilitation Robot: Mechanism Design, Human Joint Analysis and Trajectory Planning 1 , 2 , 1 1 , 3 1 4 4 Hongbo Wang *, Musong Lin , Zhennan Jin , Hao Yan , Guowei Liu , Shihe Liu and Xinyu Hu Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China; lms19910704@stumail.ysu.edu.cn (M.L.); ysdxlhx@stumail.ysu.edu.cn (Z.J.); yh@stumail.ysu.edu.cn (H.Y.) Academy for Engineering & Technology, Fudan University, Shanghai 200433, China Taiyuan locomotive depot of Daqin Railway Co., Ltd, Taiyuan 030045, China Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of Education, Yanshan University, Qinhuangdao 066004, China; lgwnn@stumail.ysu.edu.cn (G.L.); lsh@stumail.ysu.edu.cn (S.L.); huxinyu@stumail.ysu.edu.cn (X.H.) * Correspondence: hongbo_w@ysu.edu.cn; Tel.: +86-139-3366-5525 Received: 7 June 2020; Accepted: 28 June 2020; Published: 30 June 2020 Abstract: Most of currently rehabilitation robots cannot achieve the adduction/abduction (A/A) training of the hip joint and lack the consideration of the patient handling. This paper presents a four degrees of freedom (DOF) spatial workspace lower limb rehabilitation robot, and it could provide flexion/extension (F/E) training to three lower limb joints and A/A training to the hip joint. The training method is conducting the patient’s foot to complete the rehabilitation movement, and the patient could directly take training on the wheelchair and avoid frequent patient handling between the wheelchair and the rehabilitation device. Because patients own di erent joint range of motions (ROM), an analysis method for obtaining human joint motions is proposed to guarantee the patient’s joint safety in this training method. The analysis method is based on a five-bar linkage kinematic model, which includes the human lower limb. The human-robot hybrid kinematic model is analyzed according to the Denavit-Hartenberg (D-H) method, and a variable human-robot workspace based on the user is proposed. Two kinds of trajectory planning methods are introduced. The trajectory planning method and the human joint analysis method are validated through the trajectory tracking experiment of the prototype. Keywords: rehabilitation robot; human joint analysis; human-robot hybrid model; trajectory planning 1. Introduction As a common disease in the elderly population, stroke has a high probability of causing physical-motor disability [1,2]. The disability seriously a ects the lives and families of patients. There are several million newer stroke patients every year in the world [3,4]. It means that the traditional manual rehabilitation by therapists cannot meet the great demand for rehabilitations. The rehabilitation robot is an ecient human-robot interaction system, which could be applied in stroke, sport injuries and surgery rehabilitations. The e ect of rehabilitation robots has been verified and recognized through the clinical trial in last decades [5–7]. The lower rehabilitation robot has been rapidly developed in recent years, and it could be divided into exoskeletons, moving platforms and parallel platforms [8]. Exoskeletons mainly refer to the wearable human-like mechanical legs. Rehabilitation exoskeletons are generally equipped with a Appl. Sci. 2020, 10, 4542; doi:10.3390/app10134542 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 4542 2 of 17 treadmill and a weight reducing device, and exoskeletons could assist the user to complete the gait training on the treadmill [9–14]. A lower limb gait training rehabilitation robot named AIRGAIT has been developed by Shibaura Institute of Technology, and the exoskeleton of AIRGAIT mainly consists of a robotic orthosis, springs and parallel linkages [15]. Currently, the available moving platform robots have many types of mechanical structures; the training methods of them are carrying the limb to achieve the movement through terminal pedals or platforms [16–21]. A 5-DOF (degrees of freedom) hip-joint rehabilitation robot allows for full movements of the hip joint, and the user lying on the robot could be conducted to complete the training by the platform [22]. The working method of parallel platforms is same as moving platforms, and parallel platforms are generally applied in ankle joint rehabilitations because the workspace is limited by parallel structures [23–28]. A 9-DOF hybrid parallel ankle rehabilitation robot has been investigated by Rakhodaei et al., which consists of nine linear actuators and two moving platforms [29]. A comparison summary of typical lower rehabilitation robots is shown in Table 1. Table 1. Comparison summary of typical lower rehabilitation robots. Hip (H), Knee (K), Ankle (A), Leg (L), Bed (B), Wheelchair (W) and Device (D). Patient Handling Reference Training Joint Joint DOF Training Posture (Unable Stand) This paper H-K-A 2-1-1 B-W Sitting [9] H-K-A 1-1-1 B-W-D Standing [10] K 1 B-W-D Sitting [13] H-K-A 1-1-1 B-W-D Standing [14] H-K-A 1-1-1 B-W-D Standing [15] H-K-A 1-1-1 B-W-D Standing [16] H-K-A 1-1-1 B-W-D Sitting [17] H-K-A 1-1-1 B-W-D Sitting/lying [19] L 3 B-W-D Standing [18] H-K-A 1-1-1 B-W-D Sitting [20] H-K-A 1-1-1 B-W-D Sitting/lying [22] H 2 B-W-D Lying [25] A 2 B-W Sitting [26] A 3 B-W Sitting [27] A 3 B-W-D Sitting [28] L 3 B-W-D Standing [29] A 3 B-W Sitting However, although various types of rehabilitation robots have been developed, little attention is paid to the adduction/abduction (A/A) training of the hip joint, and the problem of the patient handling before training. Currently, available rehabilitation robots could seldom achieve the hip A/A training. Besides, most of rehabilitation robots are inconvenient to the patient handling. It generally takes at least two medical sta s to help the patient (no standing ability) move in the rehabilitation device. This paper proposes a 4-DOF spatial workspace rehabilitation robot, which could perform the A/A training in the hip joint and the flexion/extension (F/E) training in three joints of the lower limb. Because of the reliably mechanical structure and the newer working method, patients could directly do their rehabilitation training on wheelchairs without patient handling. On account of the spatial workspace, this robot could achieve the circumduction training of the hip joint. Circumduction training is a thigh conical motion combined with the F/E and the A/A, and several studies have proved that the circumduction training is quite helpful to the hip rehabilitation after the hip surgery [30,31]. As the working method of this robot is conducting the patient’s foot to complete the training movement, the patient’s joint range of motions (ROM) must be guaranteed. The solution is proposing an analysis method for obtaining human joint motions. This robot could plan the trajectory depending on the of patient’s joint ROM. Moreover, the research has demonstrated that there is a window of enhanced neuroplasticity early after stroke [32]. Compared to other rehabilitation robots, this robot could Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 18 an analysis method for obtaining human joint motions. This robot could plan the trajectory depending on the of patient’s joint ROM. Moreover, the research has demonstrated that there is a Appl. Sci. 2020, 10, 4542 3 of 17 window of enhanced neuroplasticity early after stroke [32]. Compared to other rehabilitation robots, this robot could participate in rehabilitation therapy earlier. That is, the patient in the flaccid paralysis participate in rehabilitation therapy earlier. That is, the patient in the flaccid paralysis period could use period could use this device under the guidance of the doctor. this device under the guidance of the doctor. In this paper, the mechanical structure of this robot is shown in the Section 1. The human-robot In this paper, the mechanical structure of this robot is shown in the Section 1. The human-robot hybrid kinematic model, the kinematic analysis and the human joint analysis are introduced in hybrid kinematic model, the kinematic analysis and the human joint analysis are introduced in Section 2. Section 2. Section 3 shows the human-robot workspace and two methods of the trajectory planning. Section 3 shows the human-robot workspace and two methods of the trajectory planning. Finally, the Finally, the trajectory tracking experiment and the result analysis are shown in the Section 4. trajectory tracking experiment and the result analysis are shown in the Section 4. 2. Mechanism Design 2. Mechanism Design This robot consists of two symmetric leg training parts, as shown in Figure 1, and each part This robot consists of two symmetric leg training parts, as shown in Figure 1, and each part includes the main motion module, the adduction/abduction motion module and the ankle motion includes the main motion module, the adduction/abduction motion module and the ankle motion module. Each leg training part could provide training for left/right legs, and the left/right leg training module. Each leg training part could provide training for left/right legs, and the left/right leg training mode could be switched by changing the assembling position of the linear actuator. This robot could mode could be switched by changing the assembling position of the linear actuator. This robot provide passive training, teaching training and resistance training. One of leg training parts is could provide passive training, teaching training and resistance training. One of leg training parts is introduced in following chapters. introduced in following chapters. Adduction/abduction motion module Ankle motion module Main motion module Figure 1. 4-DOF (degrees of freedom) spatial workspace lower limb rehabilitation robot. Figure 1. 4-DOF (degrees of freedom) spatial workspace lower limb rehabilitation robot. A As s sh shown own iin n Fi Figur gure e 2 2,, th the e m main ain m motion otion m module odule iis s a a 2 2-DOF -DOF p par ara allel llel m mechanism echanism wo working rking iin n th the e sa sagittal gittal pl plane. ane. The The22-DOF -DOF p parallel arallel m mechanism echanism h has as aa re reliably liably b bearing earing ccapability apability, , a and nd th this is d design esign co could uld re reduce duce th the e ro robot bot wi width dth si size. ze. The Thed drive rive li line ne co consists nsists oof f aa DC DCm motor otor, ,aa b ball all scre screw w aand nd aa m movable ovable ba base; se; iit t co could uld co convert nvert a angle ngle d displacements isplacements o of f th the e m motors otors iinto nto li linear near d displacements isplacements o of f ba bases. ses. T Two wo o opposite pposite d drive rive li lines nes a ar re e ffixed ixed o on n th the e un underframe. derframe. T The he co cooperation operation o of f d doub oublle e ba base se m motions otions co could uld pr pro ovide vide li linear near a and nd a angled ngled sagittal sagittal motions motionsto tothe theupper upper mechanism. mechanismThe . The range range ofo the f th angle e angl displaceme e displacem ntent is fr iom s fro 25 m 25 to°80 to ;8the 0°; th maximum e maximu linear m lindisplacement ear displacemis ent 500 is 5 mm. 00 m An m. angle An ansensor gle senis soassembled r is assembin led the in main the m link, ain land ink, a an limit d a lswitch imit sw is itch fixed is fon ixed the on far thend e farof end theomain f the m ball ain scr ba ew ll .scre The w.measuring The measuri range ng ra of nthe ge o angle f the a sensor ngle sen isso180 r is ±,1and 80°, a the nd pr thecision e precisiis on i0.2 s ±0..2The °. The sensor sensoand r and the the limit limit switch switch ar are e used used for for th the e sa safety fety d detection etection a and nd th the e a amplifier mplifier h homing. oming. The adduction/abduction motion module is a parallelogram mechanism, as shown in Figure 3a, which is fixed on the main motion module. The A/A movement of the hip is conducted by the motion of the parallelogram mechanism; the parallelogram mechanism is driven by a linear actuator assembled on the main motion module (Figure 6b). The linear actuator is in the minimum length while the parallelogram mechanism is collinear to the main motion module; the maximum angle between the parallelogram mechanism and the main motion module is 40 while the linear actuator is in the maximum length. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 18 Angle DC motor Subsidiary Main link sensor link Limit switch Appl. Sci. 2020, 10, 4542 4 of 17 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 18 Angle Subsidiary DC motor Main link sensor link Movable Ball screw Limit base switch Figure 2. Main motion module. The adduction/abduction motion module is a parallelogram mechanism, as shown in Figure 3a, which is fixed on the main motion module. The A/A movement of the hip is conducted by the motion of the parallelogram mechanism; the parallelogram mechanism is driven by a linear actuator assembled on the main motion module (Figure 6b). The linear actuator is in the minimum length while the parallelogram mechanism is collinear to the main motion module; the maximum angle Movable Ball screw between the parallelogram mechanism and the main motion module is 40° while the linear actuator base is in the maximum length. Figure 2. Main motion module. Figure 2. Main motion module. Support shaft The adduction/abduction motion module is a parallelogram mechanism, as shown in Figure 3a, Foot pedal Tension/compression which is fixed on the main motion module. The A/A movement of the hip is conducted by the motion Linear force sensor Bearing actuator of the parallelogram mechanism; the parallelogram mechanism is driven by a linear actuator module assembled on the main motion module (Figure 6b). The linear actuator is in the minimum length DC while the parallelogram mechanism is collinear to the main motion module; the maximum angle motor Support between the parallelogram mechanism and the main motion module is 40° while the linear actuator shaft is in the maximum length. Parallelogram Support mechanism Planetary shaft Sun pulley Foot pedal Tension/compression pulley Linear (a) (b) force sensor Bearing actuator module Figure 3. Adduction/abduction (a) and ankle (b) motion modules. DC Figure 3. Adduction/abduction (a) and ankle (b) motion modules. motor The structure of the ankle motion module is similar to a planetary gear train, and theSankle upportmotion shaft module could rotate around the support shaft of the parallelogram mechanism. The timing belt and The structure of the ankle motion module is similar to a planetary gear train, and the ankle pulley system is shown in Figure 3b, and it mainly consists of a sun pulley fixed on the support shaft, motion module could rotate around the support shaft of the parallelogram mechanism. The timing a planetary pulley and a timing belt working as an inner ring gear. The planetary pulley is driven by a belt and pulley system is shown in Figure 3b, and it mainly consists of a sun pulley fixed on the Parallelogram DC motor fixed under the footmec pedal. hanis The m tension/compression force sensor is assembled Planetary under the support shaft, a planetary pulley and a timing belt working as an inner ring gear. The planetary Sun pulley pedal; bearing modules fixed to the sensor are assembled on the support shaft. The measuring range pulley is driven by a DC motor fixed under the foot pedal. The tension/compression force sensor is pulley of the force sensor is5 kg. The angle displacement range of the pedal is from 60 to 60 , and it is assembled under the pe (ad ) al; bearing modules fixed to the sensor are (b a ) ssembled on the support shaft. limited by a machine key fixed on the support shaft. The measuring range of the force sensor is ±5 kg. The angle displacement range of the pedal is from −60° to 60°, and it is limited by a machine key fixed on the support shaft. Figure 3. Adduction/abduction (a) and ankle (b) motion modules. 3. Kinematic Analysis In order to obtain the human joint motions, the human lower limb is considered as a passive The structure of the ankle motion module is similar to a planetary gear train, and the ankle linkage in the kinematic model. Based on the human-robot hybrid kinematic model, the connection motion module could rotate around the support shaft of the parallelogram mechanism. The timing between the mechanism linkage and the lower limb linkage is determined. The analysis result could belt and pulley system is shown in Figure 3b, and it mainly consists of a sun pulley fixed on the be used for planning training motions by doctors, and the motions of the human joints could be shown support shaft, a planetary pulley and a timing belt working as an inner ring gear. The planetary to the doctor during the training. pulley is driven by a DC motor fixed under the foot pedal. The tension/compression force sensor is assembled under the pedal; bearing modules fixed to the sensor are assembled on the support shaft. 3.1. Human-Robot Hybrid Kinematic Model The measuring range of the force sensor is ±5 kg. The angle displacement range of the pedal is from The human-robot hybrid kinematic model could be simplified as a slider-bar linkage as shown in −60° to 60°, and it is limited by a machine key fixed on the support shaft. Figure 4. The parallel mechanism of the main motion module is equivalent to a PR mechanism (AB), and the parallelogram mechanism could be regarded as a link (BC). D, E and F represent the ankle, Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 18 3. Kinematic Analysis In order to obtain the human joint motions, the human lower limb is considered as a passive linkage in the kinematic model. Based on the human-robot hybrid kinematic model, the connection between the mechanism linkage and the lower limb linkage is determined. The analysis result could be used for planning training motions by doctors, and the motions of the human joints could be shown to the doctor during the training. 3.1. Human-Robot Hybrid Kinematic Model Appl. Sci. The 2020 hu,m 10a , n 4542 -robot hybrid kinematic model could be simplified as a slider-bar linkage as sh 5o of wn 17 in Figure 4. The parallel mechanism of the main motion module is equivalent to a PR mechanism (AB), and the parallelogram mechanism could be regarded as a link (BC). D, E and F represent the knee and hip joints of the user. As the user ’s foot is tied to the foot pedal of the ankle motion module, ankle, knee and hip joints of the user. As the user’s foot is tied to the foot pedal of the ankle motion the linear displacement extending from the ankle joint (D) to the support shaft (C) could be regarded module, the linear displacement extending from the ankle joint (D) to the support shaft (C) could be as a link (CD). The lower limb is equivalent to a passive UR mechanism (EFD), which could rotate in regarded as a link (CD). The lower limb is equivalent to a passive UR mechanism (EFD), which could the plane XOY. rotate in the plane XOY . (XZ , ) FF Z 6 O a Figure 4. Human-robot hybrid kinematic model. Figure 4. Human-robot hybrid kinematic model. The a represents the linear displacement of the movable base; , and are joint angles The a represents the linear displacement of the movable base; , and are joint angles of the 0 0 1 2 1 2 3 3 robot; and represent the A/A angle and the F/E angle of the human hip joint; represents the of the ro 5 bot; 6 and represent the A/A angle and the F/E angle of the huma7 n hip joint; 5 6 7 F/E angle of the human knee joint. is the rotation angle of l (GB as an axis), and is the rotation 2 2 5 represents the F/E angle of the human knee joint. is the rotation angle of l (GB as an axis), and 2 2 angle of l (HF as an axis). l and l are the link lengths of the robot; l is the length between the point 5 1 2 3 is the rotation angle of l (HF as an axis). l and l are the link lengths of the robot; l is the 5 5 1 2 3 C and D; l and l are the lengths of the human thigh and crus; X and Z represent the position of the 5 6 F F length between the point C and D; l and l are the lengths of the human thigh and crus; X and 5 6 F human hip in the coordinate systemfXOZg. In addition, the lower limb length of the user, the height Z represent the position of the human hip in the coordinate system XOZ . In addition, the lower from the foot sole to the ankle joint and the position of the hip should be measured before training. limb length of the user, the height from the foot sole to the ankle joint and the position of the hip 3.2. Forward/Inverse Kinematics should be measured before training. The human-robot hybrid linkage coordinate system is built as shown in Figure 5; the global 3.2. Forward/Inverse Kinematics coordinate system O x y z is located on the far end of the main ball screw while the plane x Oz is 0 0 0 0 0 parallel to the sagittal plane. The linkage chain is divided into the linkage ABC and the linkage FED The human-robot hybrid linkage coordinate system is built as shown in Figure 5; the global (right leg), and the forward/inverse kinematics of two parts are solved through the Denavit-Hartenberg coordinate system O− x y z is located on the far end of the main ball screw while the plane 0 0 0 (D-H) method in the global coordinate system O x y z . 0 0 0 As the shape change of the parallelogram mechanism cannot change the relative direction of the 1 2 opposite link, the coordinate transformation of is divided into T and T . The D-H parameters of 2 3 the linkage ABC are listed in Table 2. The directions of , and are defined against the arrow 1 3 5 directions shown in Figure 5. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 18 x Oz is parallel to the sagittal plane. The linkage chain is divided into the linkage ABC and the linkage FED (right leg), and the forward/inverse kinematics of two parts are solved through the Appl. Sci. 2020, 10, 4542 6 of 17 Denavit-Hartenberg (D-H) method in the global coordinate system O− x y z . 0 0 0 z 8 - - z 3 6 z - z x 2 x F x x 2 4 x 5 l 0 Figure 5. Human-robot hybrid linkage coordinate system. Figure 5. Human-robot hybrid linkage coordinate system. Table 2. Denavit-Hartenberg (D-H) parameters of the linkage ABC. As the shape change of the parallelogram mechanism cannot change the relative direction of the 1 2 opposite link, the coordinate tranasformatio n of is d divided into and . The D-H parameters T T i i i 1 i 1 2 2 3 of the linkage ABC are listed in Taable 2. The d 90 irections of 0 , a nd are defined against the arrow 0 1 3 1 5 l 90 0 1 2 directions shown in Figure 5. l 0 0 2 2 0 90 0 Table 2. Denavit-Hartenberg (D-H) parameters of the linkage ABC. ai−1 i 1 αi−1 di θi The transformation matrix T , which represents the coordinate transformation from frame i 1 a0 −90° 0 −θ1 to i, is obtained from Equation (1): l1 90° 0 θ2 i 1 l2 0° 0 −θ2 T = Rot(x, )Trans(x, )Rot(z, )Trans(z, d ) i 1 i 1 i i 2 3 0 −90° 0 −θ3 c s 0 a 6 i i i 1 7 6 7 6 7 6 7 6 7 s c c c s d s (1) 6 7 i i 1 i i 1 i 1 i i 1 6 7 = 6 7, i−1 6 7 The transformation m 6 atrix T , which represents the coord 7 inate transformation from frame s s c s c d c 6 i i 1i i i 1 i 1 i i 1 7 6 7 4 5 i−1 to i , is obtained from Equ 0ation (1): 0 0 1 𝑖 −1 ( ) ( ) ( ) ( ) 𝑇 = 𝑅𝑜𝑡 𝑥 ,𝛼 𝑎𝑛𝑠𝑟𝑇 𝑥 ,𝛼 𝑅𝑜𝑡 𝑧 ,𝜃 𝑟𝑇𝑎𝑛𝑠 𝑧 ,𝑑 𝑖 𝑖 −1 𝑖 −1 𝑖 𝑖 where c = cos() and s = sin(). The coordinate that transforms from x y z to x y z is calculated 0 0 0 4 4 4 𝑐 𝜃 − 𝑠 𝜃 0 𝑎 𝑖 𝑖 𝑖 −1 as follows: 2 3 (1) 𝑠 𝜃 𝑐 𝛼 𝑐 𝜃 𝑐 𝛼 − 𝑠 𝛼 − 𝑑 𝑠 𝛼 𝑖 𝑖 −1 𝑖 𝑖 −1 𝑖 −1 𝑖 𝑖 −1 =n[ o a p ] , 6 1x 1x 1x 1x 7 6 7 6 𝑠 𝜃 𝑠 𝛼 𝑐 𝜃 𝑠 𝛼 7 𝑐 𝛼 𝑑 𝑐 𝛼 𝑖 𝑖 −1 𝑖 𝑖 −1 𝑖 −1 𝑖 𝑖 −1 6 7 6 7 n o a p 6 1y 1y 1y 1y 7 0 0 1 2 3 6 7 0 0 0 1 T = T T T T = 6 7 6 7 2 3 4 1 4 6 7 n o a p 6 1z 1z 1z 1z 7 6 7 4 5 where c*= cos(*) and s*= sin(*) . The coordinate that transforms from x y z to x y z is 0 0 0 4 4 4 0 0 0 1 2 3 (2) calculated as follows: cos( + ) sin( + ) 0 l cos cos + l cos + a 6 1 3 1 3 2 1 2 1 1 0 7 6 7 6 7 6 7 6 7 0 𝑛 𝑜 0 𝑎 𝑝1 l sin 6 2 2 7 1𝑥 1𝑥 1𝑥 1𝑥 6 7 = , 6 7 6 7 6 𝑛 𝑜 𝑎 𝑝 7 sin( + ) cos( + ) 0 l sin cos + l sin 0 0 1𝑦 1𝑦 1𝑦 1𝑦 6 1 12 3 3 1 3 2 1 2 1 1 7 𝑇 = 6 𝑇 𝑇 𝑇 𝑇 = [ ] 7 4 4 1 2 3 4 5 𝑛 𝑜 𝑎 𝑝 1𝑧 1𝑧 1𝑧 1𝑧 0 0 0 1 0 0 0 1 (2) (𝜃 + 𝜃 ) (𝜃 + 𝜃 ) 0 𝑙 𝜃 𝜃 + 𝑙 𝜃 + 𝑎 where represents the1 rotation 3 angle; 1 a 3is the linear 2 sliding 1 2motion 1 of1 the0 prismatic joint. i 0 0 0 1 𝑙 𝜃 2 2 The forward/inverse kinematics of the linkage ABC could be expressed as T and Equation (3): = [ ], (𝜃 + 𝜃 ) − (𝜃 + 𝜃 ) 0 𝑙 𝜃 𝜃 + 𝑙 𝜃 1 3 1 3 2 1 2 1 1 0 0 1z0 1 > = arcsin l +l cos 1 2 2 > p > 1y = arcsin < 2 > . (3) 1z = arctan( ) 3 1 > n 1x a = p l cos l cos cos 0 1x 1 1 2 1 2 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑐𝑜𝑠 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 Appl. Sci. 2020, 10, 4542 7 of 17 Similarly, the transformation of the linkage FED could be calculated using the D-H parameters in Table 3. To analyze the linkage ABC and the linkage FED in one coordinate system, a coordinate transformation matrix T between x y z with x y z is built. The forward/inverse kinematics of 0 0 0 F F F the linkage FED is expressed as follows: 2 3 n o a p 6 2x 2x 2x 2x 7 6 7 6 7 6 7 6 7 n o a p 6 2y 2y 2y 2y 7 0 0 F 5 6 7 6 7 T = T T T T T = 6 7 6 7 8 F 5 6 7 8 6 7 n o a p 6 2z 2z 2z 2z 7 6 7 4 5 0 0 0 1 2 3 (4) ( ) cos cos + cos sin( + ) sin p 6 5 6 7 5 6 7 5 2x 7 6 7 6 7 6 7 6 7 sin cos( + ) sin sin( + ) cos p 6 5 6 7 5 6 7 5 2y 7 6 7 = 6 7, 6 7 6 7 sin( + ) cos( + ) 0 p 6 6 7 6 7 2z 7 6 7 4 5 0 0 0 1 2y = arctan( ) > 5 > X p F 2x > 2 2 2 2 < A +B +l l 5 6 B , (5) > = arccos p + arctan( ) 2 2 A > 2l A +B > 5 > B l sin 5 6 = arcsin( ) 7 6 where p = X l cos cos( + ) l cos cos 2x F 6 5 6 7 5 5 6 p = l sin cos( + ) + l sin cos 2y 6 5 6 7 5 5 6 p = Z + l sin( + ) + l sin 2z F 6 6 7 5 6 X p F 2x A = cos B = p Z 2z F Table 3. D-H parameters of the linkage FED. a d i 1 i 1 i i X 0 Z 180 F F 0 0 0 0 90 0 l 0 0 5 7 l 0 0 0 In addition, the is actually controlled by two linear displacements (a and a ) as shown in 1 0 1 Figure 6a. shown in Figure 6b is determined, and depends on the length L of the linear actuator. The linear displacements (a and L) are calculated as follows: 2 2 2 2 a = a + l cos l + l + l cos , (6) 1 0 1 1 1 4 1 1 2 2 L = l + l + 2l l sin , (7) 8 9 2 8 9 where l is the link length of the subsidiary link; l and l depend on the position of the linear actuator. 4 8 9 Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 18 2 2 2 2 (6) a = a + l cos − l + l + l cos , 1 0 1 1 1 4 1 1 L=+ l l +2l l sin (7) 8 9 8 9 2 where is the link length of the subsidiary link; and depend on the position of the linear l l l 4 8 9 Appl. Sci. 2020, 10, 4542 8 of 17 actuator. a a 0 1 (a) (b) Figure 6. Parallel mechanism model (a) and parallelogram mechanism model (b). Figure 6. Parallel mechanism model (a) and parallelogram mechanism model (b). 3.3. Analysis of Human Joints 3.3. Analysis of Human Joints The linear displacement extending from the user ankle joint to the last mechanical revolution joint The linear displacement extending from the user ankle joint to the last mechanical revolution is regarded as link l , and the human ankle is regarded as a passive spherical joint. Based on the end joint is regarded as link l , and the human ankle is regarded as a passive spherical joint. Based on point positions of the mechanism linkage ABC and the lower limb linkage FED, the connection of the the end point positions of the mechanism linkage ABC and the lower limb linkage FED, the two linkages could be built as follows: connection of the two linkages could be built as follows: p + l n = p 3 2x > 1x 1x < p + l n = p 1x 3 1x 2x p + l n = p . (8) 1y 3 1y 2y > p + l n = p . (8) 1y 3 1y 2 y p + l n = p 1z 3 1z 2z p + l n = p 1z 3 1z 2z Substituting the mechanism/human joint angular position information into Equation (8), the other Substituting the mechanism/human joint angular position information into Equation (8), the joint angular position information could be obtained. Doctors could formulate the training trajectory other joint angular position information could be obtained. Doctors could formulate the training depending on the joint ROM of patients; the motions of the patient’s joints could be calculated and trajectory depending on the joint ROM of patients; the motions of the patient’s joints could be shown to doctors during the training. calculated and shown to doctors during the training. 3.4. Velocity Analysis 3.4. Velocity Analysis The end e ector velocity can be obtained from mechanical joint velocities through the Jacobian The end effector velocity can be obtained from mechanical joint velocities through the Jacobian matrix, as in Equation (9): " # " # matrix, as in Equation (9): v . J J . l1 li = J(q) q = q, (9) ! J J a1 ai v JJ l1 li = J(q)q = q , (9) h i . . . . . . JJ a1 ai where q = a , , , , represents the joint velocities; v and ! are the linear and angular 0 1 2 2 3 velocities of the end e ector; J (linear) and J (angular) represent the velocity connections between li ai v ω where represents the joint velocities; and are the linear and angular q= a , , ,− , 0 1 2 2 3 the joint i and the end e ector. The J and J could be calculated from Equation (10): li ai J J velocities of the end effector; (linear) and (angular) represent the velocity connections li ai " # " #" # T T R R S(p ) J . v li J J between the joint i and the end effector. The and i could be calculated from Equation (10): i i li ai q = , (10) i T J 0 R ! ai TT Jv R −R S() p li i i i i q = i (10) T T i i J 0 R where R is the transpose of the rotation matrix in T ; p is the position vector in T ; v and! are linear ai i i i i i 6 6 and angular velocities in frame i; S(p ) is the skew-symmetric matrix related to p , and it is shown as: i i 2 3 6 0 p p 7 z y 6 7 6 7 6 7 6 7 S(p ) = 6 p 0 p 7. (11) z x i 6 7 6 7 4 5 p p 0 y x Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 18 T i i R T p T v where is the transpose of the rotation matrix in ; is the position vector in ; and i 6 i 6 i Appl. Sci. 2020, 10, 4542 9 of 17 S() p p are linear and angular velocities in frame i ; is the skew-symmetric matrix related to , i i i and it is shown as: . h i h i . . T T As q and q are the same variables for , J J and J J could be combined 3 4 l3 a3 l4 a4 0 −pp zy together. Finally, J(q) is calculated as: S ( p )= p 0 − p (11) i z x 2 3 cos( + ) l cos−sinpp l sin0 l sin cos 0 6 1 3 2 2 yx 3 1 3 2 2 3 7 6 7 6 7 6 7 6 7 sin( + ) l cos cos + l cos l sin sin 0 6 7 1 3 2 2 3 1 3 2 2 3 6 7 T T 6 7 6 7 q q As and are the same variables for , JJ and JJ could be combined 6 7 0 0 l cos 0 3 4 2 la 33 la 44 6 2 2 7 6 7 J(q) = 6 7. (12) 6 7 6 7 0 0 0 0 J6(q) 7 together. Finally, is calculated as: 6 7 6 7 6 7 6 7 0 0 0 0 6 7 6 7 4 5 co（ s +） −l cos sin − l sin −l sin cos 0 0 1 3 2 2 1 3 1 3 2 2 0 3 1 sin（ +） l cos cos +l cos −l sin sin 0 1 3 2 2 3 1 3 2 2 3 3.5. Kinematic Simulation of Mechanism Model 0 0 l cos 0 J(q)= (12) 0 0 0 0 To verify the kinematic equation solving, a verification based on the simulation model is conducted 0 0 0 0 in the software Automatic Dynamic Analysis of Mechanical Systems (ADAMS) as shown in Figure 7. The main steps include inputting the model, adding constraints and setting the drive equations. 0 1 0 1 h i h i The joint initial positions are a = 600 61 20 30 , and the initial position 0 2 3 of the end e ector is (1019.7, 167.6, 777). The drive equations are given as follows. 3.5. Kinematic Simulation of Mechanism Model a = 300 cos(t) + 300 > 0 To verify the kinematic equation solving, a verification based on the simulation model is = 11 cos(t) + 50 conducted in the software Automatic Dynamic Analysis of Mechanical Systems (ADAMS) as shown . (13) = 16 sin(t) + 20 in Figure 7. The main steps include i> nputting the model, adding constraints and setting the drive = 30 cos(t) 3[𝑎 𝜃 𝜃 𝜃 ] [ ] equations. The joint initial positions are = 600 61° 20° 30° , and the initial 0 1 2 3 position of the end effector is (1019.7, 167.6, 777). The drive equations are given as follows. Figure 7. Simulation model in Automatic Dynamic Analysis of Mechanical Systems (ADAMS). Figure 7. Simulation model in Automatic Dynamic Analysis of Mechanical Systems (ADAMS). After setting the other relative parameters, the displacement and velocity of the end point could 𝑎 = 300 (𝑡 )+ 300 be simulated through the software. Alternately, the end point motion information could be calculated 𝜃 = 11° (𝑡 )+ 50° { . (13) by substituting the joint information into kinematic equations. Two sets of end point motion results are 𝜃 = 16° (𝑡 )+ 20° shown in Figure 8. Comparing two curves of the end point motions, it could be found that kinematic 𝜃 = 30° (𝑡 ) calculation results are largely in agreement with simulation results from ADAMS. The calculation of After setting the other relative parameters, the displacement and velocity of the end point could the kinematic equation solving is verified. be simulated through the software. Alternately, the end point motion information could be calculated by substituting the joint information into kinematic equations. Two sets of end point motion results are shown in Figure 8. Comparing two curves of the end point motions, it could be found that 𝑐𝑜𝑠 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑐𝑜𝑠 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 18 kinematic calculation results are largely in agreement with simulation results from ADAMS. The calculation of the kinematic equation solving is verified. Appl. kinSci. ema 2020 tic ,ca 10 l,cul 4542 ation results are largely in agreement with simulation results from ADAMS. The 10 of 17 calculation of the kinematic equation solving is verified. Calculation Simulation Calculation Simulation Calculation Simulation 1050 Calculation Simulation Calculation Simulation Calculation Simulation 250 750 300 800 200 700 250 750 150 650 200 700 100 600 150 650 50 550 700 100 600 650 500 50 550 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 650 Time/s 500 Time/s Time/s 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Time/s Time/s Time/s Figure 8. Three axis comparisons of end point motion results. Figure 8. Three axis comparisons of end point motion results. Figure 8. Three axis comparisons of end point motion results. 4. Trajectory Planning 4. Trajectory Planning 4. Trajectory Planning 4.1. Human-Robot Workspace 4.1. Human-Robot Workspace 4.1. Human-Robot Workspace Before planning the training trajectory, the workspace of the robot needs to be determined. The Before planning the training trajectory, the workspace of the robot needs to be determined. Before planning the training trajectory, the workspace of the robot needs to be determined. The workspace refers to the spatial point set of the end effector, which represents the activity scope of the The workspace refers to the spatial point set of the end e ector, which represents the activity scope workspace refers to the spatial point set of the end effector, which represents the activity scope of the robots. As a virtual link (l ) is assumed between the robot and the human in the kinematic analysis robots. As a virtual link (l ) is assumed between the robot and the human in the kinematic analysis of the robots. As a virtual 3 link (l ) is assumed between the robot and the human in the kinematic of Section 3, the human ankle joint (point D) is regarded as the mechanism end point in the workspace of Section 3, the human ankle joint (point D) is regarded as the mechanism end point in the workspace analysis of Section 3, the human ankle joint (point D) is regarded as the mechanism end point in the analysis. The workspace is analyzed with the numerical method, and the robot workspace is a analysis. The workspace is analyzed with the numerical method, and the robot workspace is a workspace analysis. The workspace is analyzed with the numerical method, and the robot workspace hexahedron as shown in Figure 9. Substituting the extreme value of each joint into the forward hexahedron as shown in Figure 9. Substituting the extreme value of each joint into the forward is a hexahedron as shown in Figure 9. Substituting the extreme value of each joint into the forward kinematic, and workspace boundaries of the robot could be determined. kinematic, and workspace boundaries of the robot could be determined. kinematic, and workspace boundaries of the robot could be determined. 0mm a 500mm 0mm a 500mm 600600 25 25 65 65 1 1 0 40 0 40 2 2 400 -60 60 -60 60 X/mm 1200 X/mm 1200 0 200 Y/mm Y/mm Figure 9. The spatial workspace of the robot. Figure 9. The spatial workspace of the robot. Figure 9. The spatial workspace of the robot. In effect, the robot workspace could not directly be applied for the training planning due to the In e ect, the robot workspace could not directly be applied for the training planning due to the limitedIn ra ef nge fecof t, th hum e ro an bo jo t in wo t m rkspa otion ce s. co The uld in n ter ot d secti ire o ct nl y ofb th e e appl robio etd wo for rkspa the tra ce i an nid n g thp e la lo nwer ning li d m ue b to the limited range of human joint motions. The intersection of the robot workspace and the lower limb mlo im tioin ted spa ra ce n ge is fof ea si hb um le f ao n r jth oin e t tra m in oiti no gn pl s.a The nninig. nter Besi secti des, o n pa oti f ents the ro wib th o td wo iffer rkspa ent lice m b a n lengt d thh e s la on wer d limb motion space is feasible for the training planning. Besides, patients with di erent limb lengths and joint ROM own different motion spaces; therefore, the stable workspace is not suitable for each motion space is feasible for the training planning. Besides, patients with different limb lengths and joint ROM own di erent motion spaces; therefore, the stable workspace is not suitable for each patient. joint ROM own different motion spaces; therefore, the stable workspace is not suitable for each To guarantee the patient’s safety and to avoid secondary damage, a variable human-robot workspace is proposed. The variable human-robot workspace is the overlapping part of two spaces, and it is changed depending on different parameters of the human limb. Figure 10 shows a situation of the sagittal training, Displacement in X axis/mm Displacement in X axis/mm Z/mm Z/mm Displacement in Y axis/mm Displacement in Y axis/mm Displacement in Z axis/mm Displacement in Z axis/mm Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 18 patient. To guarantee the patient’s safety and to avoid secondary damage, a variable human-robot workspace is proposed. Appl. Sci. 2020, 10, 4542 11 of 17 The variable human-robot workspace is the overlapping part of two spaces, and it is changed depending on different parameters of the human limb. Figure 10 shows a situation of the sagittal and t tra hein via nr g, iaa bn le d h th u e m va an ri- arb olb e o htum wo ar nk -ro spb ao cte wo is t rksp he sa h ce ad is oth we p sh ara td . o Tw he ph art. ip The joint hi p p ojs oiin tio t n po s si (X tion as n(dXZ an )d a re set F F Z ) are set as 1790 mm and 600 mm. The lengths of the human thigh (l ) and crus (l ) are 396 mm and F 5 6 as 1790 mm and 600 mm. The lengths of the human thigh (l ) and crus (l ) are 396 mm and 496 mm. 5 6 496 mm. -0 0 1200 -10 60 -110 0 200 400 600 800 1000 1200 1400 1600 X/mm Figure 10. The variable human-robot workspace. Figure 10. The variable human-robot workspace. 4.2. Trajectory Planning Mehthod 4.2. Trajectory Planning Mehthod Two methods for the trajectory planning are presented in this section. One is formulating the Two methods for the trajectory planning are presented in this section. One is formulating the trajectory in the human-robot workspace directly, and then calculating the mechanical joint motions trajectory in the human-robot workspace directly, and then calculating the mechanical joint motions through the inverse kinematic. The other is inputting the patient’s joint information, including the through the inverse kinematic. The other is inputting the patient’s joint information, including the ROM and the training speed; the next step is constructing the trajectory and then calculating the ROM and the training speed; the next step is constructing the trajectory and then calculating the mechanical joint motions. Two kinds of the training trajectories planned by two methods are shown mechanical joint motions. Two kinds of the training trajectories planned by two methods are shown in in Figures 11 and 12. Figures 11 and 12. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18 The circle training trajectory is applied in the sagittal multiple joint training, which is calculated by the first method. The main approach is calculating the typical incircles of the workspace boundaries, and other circles could be obtained by modifying the typical circles. The position of the incircle center could be obtained from Equation (14), and the circle trajectory is expressed as Equation (15). 2 2 2 (x − x ) + (z − z ) = (r + r ) 0 1 0 1 1 0 1000 , (14) 2 2 2 (x − x ) + (z − z ) = (r − r ) 0 2 0 2 2 0 x=+ x r sin(t ) (15) z=+ z r cos(t ) 00 200 400 600 800 1000 1200 1400 1600 X/mm Figure 11. Circle training trajectory. Figure 11. Circle training trajectory. where x and z represent the center position of the trajectory circle; r is the radius of the 0 0 0 trajectory circle. x and z (𝑖 = 1 and 2) represent the center position of boundary arcs; r (𝑖 = i i i 1 and 2) represents the radius of boundary arcs. represents the angular velocity of the trajectory circle, and represents time. Because there are multiple inverse position solutions of the robot mechanism model, the single position planning cannot meet the requirement for all joints. Therefore, it is necessary to add the angle ( n,, o a ) planning of the end effector. The main principle is remaining the absolute angle 1 1 1 value of the end effector in a lower degree to guarantee the human joints are trained in safe range. X/mm Y/mm Figure 12. Spherical ellipse training trajectory. The spherical ellipse training trajectory is used for the hip circumduction rehabilitation, and it is planned depending on the patient’s joint ROM through the second method. To guarantee the training effect of the hip joint, the patient’s leg is left straight in the circumduction training. The rotation center Z/mm Z/mm Z/mm Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18 200 400 600 800 1000 1200 1400 1600 X/mm Figure 11. Circle training trajectory. where x and z represent the center position of the trajectory circle; r is the radius of the 0 0 0 trajectory circle. x and z (𝑖 = 1 and 2) represent the center position of boundary arcs; r (𝑖 = i i i 1 and 2) represents the radius of boundary arcs. represents the angular velocity of the trajectory circle, and t represents time. Because there are multiple inverse position solutions of the robot mechanism model, the single position planning cannot meet the requirement for all joints. Therefore, it is necessary to add the angle (n,, o a ) planning of the end effector. The main principle is remaining the absolute angle Appl. Sci. 2020, 101, 4542 1 1 12 of 17 value of the end effector in a lower degree to guarantee the human joints are trained in safe range. X/mm Y/mm Figure 12. Spherical ellipse training trajectory. Figure 12. Spherical ellipse training trajectory. The spherical ellipse training trajectory is used for the hip circumduction rehabilitation, and it is The circle training trajectory is applied in the sagittal multiple joint training, which is calculated planned depending on the patient’s joint ROM through the second method. To guarantee the training by the first method. The main approach is calculating the typical incircles of the workspace boundaries, effect of the hip joint, the patient’s leg is left straight in the circumduction training. The rotation center and other circles could be obtained by modifying the typical circles. The position of the incircle center could be obtained from Equation (14), and the circle trajectory is expressed as Equation (15). 2 2 2 (x x ) + (z z ) = (r + r ) 0 1 0 1 1 0 , (14) 2 2 2 (x x ) + (z z ) = (r r ) 0 2 0 2 2 0 x = x + r sin(!t) 0 0 , (15) z = z + r cos(!t) 0 0 where x and z represent the center position of the trajectory circle; r is the radius of the trajectory 0 0 0 circle. x and z (i = 1 and 2) represent the center position of boundary arcs; r (i = 1 and 2) represents the i i i radius of boundary arcs. ! represents the angular velocity of the trajectory circle, and t represents time. Because there are multiple inverse position solutions of the robot mechanism model, the single position planning cannot meet the requirement for all joints. Therefore, it is necessary to add the angle ([n , o , a ]) planning of the end e ector. The main principle is remaining the absolute angle value of 1 1 1 the end e ector in a lower degree to guarantee the human joints are trained in safe range. The spherical ellipse training trajectory is used for the hip circumduction rehabilitation, and it is planned depending on the patient’s joint ROM through the second method. To guarantee the training e ect of the hip joint, the patient’s leg is left straight in the circumduction training. The rotation center could be calculated from the simplified Equation (4) by inputting the joint angle ( and ) ranges of 5 6 the hip joint, and the trajectory of the human ankle could be built as follows: > x = X (l + l )cos cos 0 F 6 5 5 6 y = (l + l )sin cos , (16) 0 6 5 5 6 z = Z + (l + l )sin 0 F 6 5 6 > x = X (l + l )cos( + Asin(!t))cos( + Bcos(!t)) F 6 5 5 6 , (17) > y = (l + l )sin( + Asin(!t))cos( + Bcos(!t)) 6 5 5 6 z = Z + (l + l )sin( + Bcos(!t)) F 6 5 6 where x , y and z represent the rotation center position of the trajectory; represents the average 0 0 0 i value of the joint angle range. A and B refer to the major semi-axis and minor semi-axis of the ellipse trajectory, which are related to joint angle ranges. Z/mm Z/mm Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18 could be calculated from the simplified Equation (4) by inputting the joint angle ( and ) ranges 5 6 of the hip joint, and the trajectory of the human ankle could be built as follows: x =X −+ (l l ) cos cos 0 F 6 5 5 6 y =(l + l ) sin cos , (16) 0 6 5 5 6 z =Z ++ (l l ) sin 0 F 6 5 6 x=X − (l + l ) cos( + A sin(t )) cos( + B cos(t )) F 6 5 5 6 y=(l + l ) sin( + A sin(t )) cos( + B cos(t )) , (17) 6 5 5 6 z=Z + (l + l ) sin( + B cos( t)) F 6 5 6 where x , y and z represent the rotation center position of the trajectory; represents the 0 0 0 i Appl. Sci. 2020, 10, 4542 13 of 17 average value of the joint angle range. A and B refer to the major semi-axis and minor semi-axis of the ellipse trajectory, which are related to joint angle ranges. A As s th the e seco second nd m method ethod ffirstly irstly co considers nsiders th the e h human uman jjoint oint ra ranges, nges, tthe he f final inal tra trajectory jectory sh should ould b be e checked and modified to avoid exceeding the robot workspace. Each final trajectory calculated from checked and modified to avoid exceeding the robot workspace. Each final trajectory calculated from tw two o m methods ethods is is d divided ivided iinto nto a a set set of of m multitu ultitude de poi points nts b by y t the he numerical numerical m method. ethod. The The po position sition o of f th the e trajectory points could be obtained directly from the trajectory analytical formula; the velocity and trajectory points could be obtained directly from the trajectory analytical formula; the velocity and a acceleration cceleration between betweenevery every point poin ar t eaplanned re plannto edmake to mthe ake speed the ssmooth. peed sm The ooth mechanical . The mech joint anica motions l joint motions could be calculated from the position array of points by inverse solving. Then, the motor could be calculated from the position array of points by inverse solving. Then, the motor control control commands could be determined from mechanical joint motions. The semi-close loop position commands could be determined from mechanical joint motions. The semi-close loop position control control is selected in the trajectory tracking experiment, and it is more suitable than other controls in is selected in the trajectory tracking experiment, and it is more suitable than other controls in this this accurate trajectory training. accurate trajectory training. To guarantee the patient’s safety, a human joint check function is built. It could verify and To guarantee the patient’s safety, a human joint check function is built. It could verify and display display the human joint angle through the kinematic calculations. If the angle exceeds the preset or the human joint angle through the kinematic calculations. If the angle exceeds the preset or limitation limitation value, it would stop the training to avoid the secondary damage. Training trajectories value, it would stop the training to avoid the secondary damage. Training trajectories provided by this provided by this robot include the circle, straight, curve, helix and other spatial trajectories. Because robot include the circle, straight, curve, helix and other spatial trajectories. Because the methods are the methods are almost same, no more details are shown in this section. almost same, no more details are shown in this section. 5. Prototype Experiment 5. Prototype Experiment In this section, trajectory tracking experiments were conducted to verify the trajectory planning and In this section, trajectory tracking experiments were conducted to verify the trajectory planning the human joint analysis method. A healthy subject was selected to associate the experiment, and the and the human joint analysis method. A healthy subject was selected to associate the experiment, and informed consent was confirmed and signed by the subject before the experiment. The prototype of the informed consent was confirmed and signed by the subject before the experiment. The prototype this robot and relative parameters are shown in Figure 13. of this robot and relative parameters are shown in Figure 13. Gender male Height 170 cm Thigh 470 mm Crus 377 mm Hip A/A 0°~30° Hip F/E 0°~80° Knee 120°~0° Hip position (1790, 600) Figure 13. Prototype of the robot and relative experiment parameters. Figure 13. Prototype of the robot and relative experiment parameters. The subject parameters were inputted into the control system, and the adaptive workspace could The subject parameters were inputted into the control system, and the adaptive workspace could be created. The right leg of the subject was remained relaxed, and the right foot was fixed to the robot be created. The right leg of the subject was remained relaxed, and the right foot was fixed to the robot pedal. Two 3-axis absolute angle sensors were tied to the thigh and crus of the subject, which were used for detecting angle displacements of subject joints. Two passive trainings including the circle training and spherical ellipse training were conducted in this experiment. Each training was run for five cycles; the recorded data were taken for average processing and then were drawn into figures by MATLAB. The circle trajectory was modified based on the maximum incircle of the S1, S2 and S3 boundary arcs, which could be described through Equations (14) and (15). The circle center position was (1108.3, 0, 508.7), and the radius was 130 mm. The angular velocity ! was set as 0.11 rad/s. The end point position theoretical calculation and the human joint angle theoretical calculation are shown in Figures 14 and 15. The actual positions of the end point could be obtained through angle sensors and encoders, and the actual angles of the human joints were detected by sensors tied to the subject’s leg. Experimental results are also shown in Figures 14 and 15. The end point maximum displacement errors between the calculation and the experiment were 5.46 mm and 4.84 mm in the X- and Z-axes, Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18 pedal. Two 3-axis absolute angle sensors were tied to the thigh and crus of the subject, which were pedal. Two 3-axis absolute angle sensors were tied to the thigh and crus of the subject, which were used for detecting angle displacements of subject joints. Two passive trainings including the circle used for detecting angle displacements of subject joints. Two passive trainings including the circle training and spherical ellipse training were conducted in this experiment. Each training was run for training and spherical ellipse training were conducted in this experiment. Each training was run for five cycles; the recorded data were taken for average processing and then were drawn into figures by five cycles; the recorded data were taken for average processing and then were drawn into figures by MATLAB. MATLAB. The circle trajectory was modified based on the maximum incircle of the S1, S2 and S3 boundary The circle trajectory was modified based on the maximum incircle of the S1, S2 and S3 boundary arcs, which could be described through Equations (14) and (15). The circle center position was (1108.3, arcs, which could be described through Equations (14) and (15). The circle center position was (1108.3, 0, 508.7), and the radius was 130 mm. The angular velocity was set as 0.11π rad/s. The end point 0, 508.7), and the radius was 130 mm. The angular velocity was set as 0.11π rad/s. The end point position theoretical calculation and the human joint angle theoretical calculation are shown in Figures position theoretical calculation and the human joint angle theoretical calculation are shown in Figures 14 and 15. The actual positions of the end point could be obtained through angle sensors and 14 and 15. The actual positions of the end point could be obtained through angle sensors and encoders, and the actual angles of the human joints were detected by sensors tied to the subject’s leg. encoders, and the actual angles of the human joints were detected by sensors tied to the subject’s leg. Appl. Sci. 2020, 10, 4542 14 of 17 Experimental results are also shown in Figures 14 and 15. The end point maximum displacement Experimental results are also shown in Figures 14 and 15. The end point maximum displacement errors between the calculation and the experiment were 5.46 mm and 4.84 mm in the X- and Z-axes, errors between the calculation and the experiment were 5.46 mm and 4.84 mm in the X- and Z-axes, and the human joint angle maximum displacement errors between the calculation and the experiment and the human joint angle maximum displacement errors between the calculation and the experiment and the human joint angle maximum displacement errors between the calculation and the experiment were 1.45° and 1.99° in and . were 1.45° and 1.99° in and . 6 7 were 1.45 and 1.99 in and 6 . 7 6 7 Calculation displacement in X Calculation displacement in X Calculation displacement in Z Calculation displacement in Z Calculation displacement in Y Calculation displacement in Y Actual displacement in X Actual displacement in X Actual displacement in Z Actual displacement in Z -100 -100 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Time/s Time/s Figure 14. Comparisons of displacements between the theoretical and the experiment. Figure 14. Comparisons of displacements between the theoretical and the experiment. Figure 14. Comparisons of displacements between the theoretical and the experiment. Calculation angle in θ Calculation angle in θ 5 Calculation angle in θ Calculation angle in θ Calculation angle in θ Calculation angle in θ Actual angle in θ Actual angle in θ 6 -25 Actual angle in θ -25 7 Actual angle in θ -50 -50 -75 -75 -100 -100 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Time/s Time/s Figure 15. Comparisons of angles between the theoretical and the experiment. The spherical ellipse trajectory is planned depending on the training requirement and the patient’s ROM, and it is mainly used for the circumduction training. The spherical ellipse trajectory could be described through Equations (16) and (17). The training range of the hip A/A was set from 0 to 20 ; the range of the hip F/E was set from 5 to 5 ; the angular velocity ! was set as 0.11 rad/s. The comparisons of the end point positions and human joint angles are shown in Figures 16 and 17. The end point maximum displacement errors between the calculation and the experiment were 3.01 mm, 5.46 mm and 4.19 mm in the X-, Y- and Z-axes, and the human joint angle maximum displacement errors between the calculation and the experiment were 0.94 and 1.15 in and . 5 6 Displacement/mm Angle/degree Displacement/mm Angle/degree A Ap pp pll.. S Sccii.. 20202020, , 10 10,, x x F FO OR R P PEE EER R R RE EV VIIEW EW 15 15 o off 18 18 Figure 15. Comparisons of angles between the theoretical and the experiment. Figure 15. Comparisons of angles between the theoretical and the experiment. The The s spher pheriica call el elli lipse pse tra trajjec ecto tory ry iis s pl pla an nn ned ed d dep ependi endin ng g on on th the e tr tra aiin niin ng g re req qui uire rem ment ent a an nd d tth he e pa pati tient ent’s ’s R ROM OM,, a an nd d iit t iis s m ma aiin nlly y used used ffo or r th the e c ciirc rcum umd ducti uctio on n tra traiin niin ng g.. The The s sp ph her eriica call elli ellipse pse tra trajjec ecto tory ry could be described through Equations (16) and (17). The training range of the hip A/A was set from could be described through Equations (16) and (17). The training range of the hip A/A was set from 0° to 20°; the range of the hip F/E was set from −5° to 5°; the angular velocity was set as 0.11π 0° to 20°; the range of the hip F/E was set from −5° to 5°; the angular velocity was set as 0.11π rad/s. The comparisons of the end point positions and human joint angles are shown in Figures 16 rad/s. The comparisons of the end point positions and human joint angles are shown in Figures 16 and 17. The end point maximum displacement errors between the calculation and the experiment and 17. The end point maximum displacement errors between the calculation and the experiment were 3.01 mm, 5.46 mm and 4.19 mm in the X-, Y- and Z-axes, and the human joint angle maximum Appl. Sci. were 2020 3 , .10 01 , 4542 mm, 5.46 mm and 4.19 mm in the X-, Y- and Z-axes, and the human joint angle maximum 15 of 17 d diispl spla ace cem ment ent er erro rors rs b betwee etween n th the e ca callcul cula ati tio on n a an nd d tth he e ex expe peri rim ment ent were were 0 0..9 94 4° ° a an nd d 1 1..1 15 5° ° iin n a an nd d .. 5 6 5 6 900 900 800 800 500 500 Calculation displacement in X Calculation displacement in X Calculation displacement in Z Calculation displacement in Z 400 400 Ca Callc cu ulla attiio on n d diissp plla ac ce em me en ntt iin n YY Actual displacement in X Actual displacement in X 300 300 Actual displacement in Z Actual displacement in Z A Ac cttu ua all d diissp plla ac ce em me en ntt iin n YY 100 100 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Time/s Time/s Figure 16. Comparisons of displacements between the theoretical and the experiment. Fig Figure ure 1 16 6.. Co Com mpa parris iso on ns s o of f di displ spla ac cemen ementts s b bet etw ween een tth he e tth heo eorret etica ical l a an nd d tth he e ex expe perrimen imentt.. Calculation angle in θ Calculation angle in θ5 20 Calculation angle in θ 20 Calculation angle in θ 6 Ca Callc cu ulla attiio on n a an ng glle e iin n θθ 7 7 Actual angle in θ Actual angle in θ 5 15 Actual angle in θ Actual angle in θ6 -5 -5 -10 -10 00 22 44 66 88 10 10 12 12 14 14 16 16 18 18 Time/s Time/s Figure 17. Comparisons of angles between the theoretical and the experiment. Figure Fig 17. ureComparisons 17. Comparison of s o angles f anglebetween s betweenthe the theor theoret etical ical aand nd th the e ex experiment. periment. 6. Discussion 6 6.. D Diisc scu uss ssiion on Figures 14 and 16 show that the trajectory errors between the calculation and the experiment are at a low level, so the result could prove that this robot has a great capability and could provide accurate trajectory motions. From Figures 15 and 17, it could be found that the actual human joint motions have the same pattern to theoretical calculations. Compared to manual rehabilitation, the error is in the acceptable range. The experiment results indicate that this robot has a good performance in hip A/A and sagittal trainings. This robot could acquire the valid information of patient’s joint motions in training, and doctors could design the training depending on the patient’s joint ROM through the robot. Therefore, it is feasible to regard this robot as an alternative solution to the traditional lower rehabilitation. Rehabilitation medicine is a wide subject, and it is mainly divided into neurological rehabilitation (stroke) and orthopedic rehabilitation (surgery). There are both similarities and di erences between the two kinds of rehabilitations, and this device mainly targets stroke patients to help the patient in avoiding limb physical-motor disability. In the future, more training functions for stroke would be studied based on clinical applications. Displacement/mm Displacement/mm Angle/degree Angle/degree Appl. Sci. 2020, 10, 4542 16 of 17 In the kinematic model of the lower limb, the ankle joint motion is not fully considered. This issue is regarded as the main source of human joint errors. Meanwhile, the behavior of the A/A training is not very well when the leg is not straight. The little axial rotation of the leg is the main reason of this situation. These problems would be investigated in the future work. 7. Conclusions A 4-DOF serial-parallel hybrid lower limb rehabilitation robot with the spatial workspace is introduced in this article. The mechanism characters of this robot are the simple structure and the small size, and the patient could directly do training from a wheelchair without patient handling. The training movements of this robot include the hip A/A movement and F/E movements of three lower joints. To guarantee the joint ROM, a method for acquiring the human joint motions is proposed. This analysis method is based on a human-robot hybrid kinematic model. The joint motion information could be used in the training detection and the trajectory planning. Two kinds of trajectory planning methods in a variable human-robot workspace are introduced. Finally, the trajectory tracking experiment of the prototype approves the accuracy of the robot trajectory planning and the feasibility of the human joint analysis method. This robot could be a low-cost alternative solution for manual rehabilitation because of the capability of training behaviors, and it has a good potential to be applied in hospitals or nursing homes. Author Contributions: Conceptualization, H.W. and M.L.; methodology, Z.J. and H.Y.; software, X.H.; prototype and experiment, G.L. and S.L.; writing—original draft preparation, M.L.; writing—review and editing, H.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National key research and development program (2019YFB1312500), National Natural Science Foundation of China (U1913216), Key research and development plan of Hebei Province, China (19211820D), Shanghai Science and Technology Innovation Action Plan (19441908200). Conflicts of Interest: The authors declare no conflict of interest. References 1. Tyson, S.F.; Hanley, M.; Chillala, J.; Selley, A.; Tallis, R.C. Balance disability after stroke. Phys. Ther. 2006, 86, 30–38. [CrossRef] [PubMed] 2. Munyombwe, T.; Hill, K.M.; Knapp, P.; West, R.M. Mixture modelling analysis of one-month disability after stroke: Stroke outcomes study (SOS1). Qual. Life Res. 2014, 23, 2267–2275. [CrossRef] 3. Zhang, X.; Yue, Z.; Wang, J. Robotics in Lower-Limb Rehabilitation after Stroke. Behav. Neurol. 2017, 2017, 1–13. [CrossRef] 4. Ochi, M.; Wada, F.; Saeki, S.; Hachisuka, K. 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Journal
Applied Sciences – Multidisciplinary Digital Publishing Institute
Published: Jun 30, 2020