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Conceptual Design and Computational Modeling Analysis of a Single-Leg System of a Quadruped Bionic Horse Robot Driven by a Cam-Linkage Mechanism

Conceptual Design and Computational Modeling Analysis of a Single-Leg System of a Quadruped... Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 2161038, 13 pages https://doi.org/10.1155/2019/2161038 Research Article Conceptual Design and Computational Modeling Analysis of a Single-Leg System of a Quadruped Bionic Horse Robot Driven by a Cam-Linkage Mechanism Liangwen Wang , Weiwei Zhang, Caidong Wang , Fannian Meng , Wenliao Du , and Tuanhui Wang School of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou 450002, China Correspondence should be addressed to Liangwen Wang; w_liangwen@sina.com and Caidong Wang; vwangcaidong@163.com Received 24 May 2019; Accepted 3 October 2019; Published 4 November 2019 Academic Editor: Alberto Borboni Copyright © 2019 Liangwen 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. In this study, the configuration of a bionic horse robot for equine-assisted therapy is presented. A single-leg system with two degrees of freedom (DOFs) is driven by a cam-linkage mechanism, and it can adjust the span and height of the leg end-point trajectory. After a brief introduction on the quadruped bionic horse robot, the structure and working principle of a single-leg system are discussed in detail. Kinematic analysis of a single-leg system is conducted, and the relationships between the structural parameters and leg trajectory are obtained. On this basis, the pressure angle characteristics of the cam-linkage mechanism are studied, and the leg end-point trajectories of the robot are obtained for several inclination angles controlled by the rotation of the motor for the stride length adjusting. The closed-loop vector method is used for the kinematic analysis, and the motion analysis system is developed in MATLAB software. The motion analysis results are verified by a three-dimensional simulation model developed in Solidworks software. The presented research on the configuration, kinematic modeling, and pressure angle characteristics of the bionic horse robot lays the foundation for subsequent research on the practical application of the proposed bionic horse robot. 1. Introduction researchers have extensively studied the structure, move- ment, and control of the robot legs. Li et al. [5] systemati- The quadruped walking robots are an important type of cally studied a single-leg system of a quadruped bionic robot. Based on the analysis of the muscle-bone structure legged robots. Recently, many countries have conducted profound research on the walking robots, which denote a of quadrupeds, the DOF configuration of a single leg was frontier technology of the strategic significance. The BigDog determined. Chen et al. [6] studied a bionic quadruped robot developed by Boston Dynamics is a rough-terrain robot that in order to improve its dynamic stability and adaptability by captures the mobility, autonomy, and speed of living crea- imitating the quadrupeds. Chen et al. [7] designed a new tures, which is a typical example of a legged robot [1, 2]. bionic robot named Hound. The body structure, especially The cheetah robots presented in [3] are capable of many that of the legs, and geometric relationships of the Hound functions, among which are sprinting and sharp turning, robot were designed based on the bionic research. Anantha- which is similar to the kinematics of the biological proto- narayanan et al. [8] designed the bionic legs suitable for types. The quadruped walking robots are equipped with high-speed motion, taking into consideration the balance multibranched motion mechanisms. The multi-degree-of- between weight and strength. Smith and Jivraj [9] compared freedom coupling between the branches makes the coordina- a hind leg with a leg with a different arrangement, demon- tion of robot motion very complicated [4]. In order to strating in detail how leg orientation can affect the dynamic improve the mobility and load capacity of a robot, several characteristics and gait performance of a robot. Seok et al. 2 Applied Bionics and Biomechanics tions for the double-dwell follower were used, basically syn- [10] introduced the design principles of the highly efficient walking robots, which were implemented in the design and thesized applying the cycloidal and polynomial motion experimental analysis of a cheetah robot. curves to a monomial basis [26]. Rothbart [27] presented a constant-breadth circular-arc cam profile in which the fol- Nowadays, it is relatively common to use robots in human rehabilitation and treatment processes. The robotized lower movement denoted a double-dwell function. Qian systems have been used for rehabilitation, improving the [28] investigated a constant-diameter cam mechanism, which efficacy while reducing the healthcare costs [11]. The reha- included a double roller follower with a planar movement that bilitation robots have significant advantages in helping was joined to an output rocker arm, and used a graphical- analytical method to establish the relationship between the patients to recover from neurological disorders and to become able to walk again [12]. Over the last few decades, geometric parameters of the designed constant-diameter many kinds of lower limb rehabilitation robots have been cam mechanism and the cam angles. developed [13–15]; however, their high cost makes them The circular-arc cams are easy to design, manufacture, impracticable for home healthcare. and test, which makes them cheaper than others [29, 30]. Cardona et al. [31] studied two constant-breadth cam mech- Recently, several one-DOF mechanisms have been pro- posed for a rehabilitation process, such as the four-bar anisms and presented the equations for calculating the cam mechanisms proposed by Alves et al. [16], the Stephenson breadth when the translating follower is eccentric with an III six-bar mechanism [17], and the ten-bar linkage mecha- inclination, the radius of curvature of its profile and the slid- nism proposed by Tsuge and Mccarthy [18]. With the aim ing velocities of constant-breadth cam mechanisms with translating and oscillating followers. to overcome the limitation of such mechanisms that the desired gait path can be matched only approximately, Kay In the proposed constant-breadth three-center cam and Haws [19] proposed a cam-linkage mechanism, combin- design, a swing center point of the follower can slide along ing a fixed cam and a four-bar linkage to generate the desired the guide way, resulting in a complex plane follower motion. path accurately. Mundo et al. [20] synthesized the cam- Recently, there have been a few reports on the cam-linkage mechanism where a constant-breadth three-center cam was linkage mechanisms with one or more cams for precise path generation, while Soong [21] proposed a novel cam-geared used [31]. mechanism for path generation. Shao et al. [22] designed From the aspects of the overall robot structure, the struc- ture and the working principle of a single-leg system of the a new robot structure, where a seven-bar crank-slider mech- anism was combined with a cam to generate a precise target robot driven by a cam-linkage mechanism are discussed in detail. The kinematics modeling and analysis of the cam- path. The novel lower-limb rehabilitation system was com- posed of a body weight support system to unload the body linkage mechanism are carried out, and the relationships weight and two cam-linkage mechanisms to generate the between the structural parameters and foot trajectory of a single leg are obtained. On this basis, the pressure angle char- natural gait trajectory and guide the feet of a patient. As an unconventional therapy, an equine-assisted ther- acteristics of the cam-linkage mechanism are thoroughly investigated. The closed-loop vector method is used for the apy provides horse riding training [23], which allows the user’s pelvis and torso to feel the movement of the horse in kinematic analysis, and MATLAB software is used for the order to improve balance control, promote trunk extension, development of the motion analysis system. Moreover, the trajectory cluster of a robot meeting the pressure angle condi- make a rhythmic rotation of the trunk, and enhance endur- ance performance and cardiopulmonary function [24, 25]. tions is obtained. The motion analysis results are verified by the simulation with a three-dimensional model established At present, it is costly to use horses for equine-assisted ther- apy, so equestrian therapy is used as a relatively less-costly in Solidworks software. solution. In this study, in order to imitate the movement of The rest of the paper is organized as follows. In Section 2, the structure of the bionic horse robot driven by a cam- a horse and achieve effective equine-assisted therapy, a bionic horse robot driven by a cam-linkage mechanism is designed. linkage mechanism is briefly introduced. The working prin- Each leg of the proposed bionic horse robot has two driv- ciple of the proposed robot is explained in detail in Section 3. ing motors, which can adjust the span and height of the leg In Section 4, the proposed cam-linkage mechanism is pre- motion path. In the normal walking by using the leg motion, sented and analyzed. In Section 5, the pressure angle charac- teristics of the proposed robot are studied. The calculation only one driving motor is needed. The designed cam-linkage mechanism uses a constant- example, the motion simulation, and the obtained results breadth three-center cam. The constant-breadth cam mech- are given in Section 6. Finally, the conclusions are drawn anism belongs to the class of desmodromic or positive drive in Section 7. mechanisms. In the case of a parallel flat-faced double follower, the distance of contact points between the two 2. Structure of a Bionic Horse Robot Driven by a flat-faced followers and cam denotes the cam breadth. The Cam-Linkage Mechanism constant-breadth cams may be the circular-arc cams or hav- ing an arbitrary geometry and can operate either translating Following the bionic theory, a bionic horse robot imitates or oscillating a follower if the appropriate desmodromic con- the movement of a horse. A single-leg motion of a bionic ditions are established. horse robot is driven by a cam-linkage mechanism. The In the studies on the constant-breadth cam mechanisms, overall structure of the robot is presented in Figure 1, usually, the circular-arc cam profiles and displacement func- where it can be seen that the robot consists of the robot Applied Bionics and Biomechanics 3 301 302 303 304 305 307 308 323 311 324 312 326 314 327 315 328 316 329 317 330 318 331 319 1 2 3 332 320 Figure 1: The structure of the bionic horse robot. body (denoted by 1), the control system (denoted by 2), and the single-leg walking system (denoted by 3). The developed robot had four parallel and symmetrical single-leg walking Figure 2: The structure of the single-leg walking system. systems placed on both sides of the robot body. The structure of the single-leg walking system and the schematic of the single-leg walking mechanism are shown in Figures 2 and point W. During walking, driven by a driving motor 3, respectively. for the stride length (303), astride length fork (305) In Figures 2 and 3, 301 denotes the connection plate, 302 moves and swings forward or backward along with denotes the connection bolt, 303 denotes the driving motor the rotation of astride length cam. The motion of for the stride length, 304 denotes the drive shaft of the stride the stride length fork makes a connecting rod (316) length cam, 305 denotes the stride length fork, 306 denotes to rotate, thus moving the walking leg (319) forward the stride length cam, 307 denotes the fork connecting rod, or backward 308 denotes the rectangular swing rod, 309 denotes the short (2) The stride length adjusting mechanism, defined by slide block, 310 denotes the connecting pin shaft I, 311 the path O O O O O in Figure 3, mainly controls 1 7 3 2 1 denotes the lead screw, 312 denotes the coupler, 313 denotes the horizontal movement of the leg end-point W.A the body connecting pin shaft I, 314 denotes the connecting motor for the stride length adjusting (328) is con- pin shaft II, 315 denotes the stride height cam, 316 denotes nected with a lead screw (311) through a coupler. the connecting rod, 317 denotes the connecting pin shaft Rotation of the lead screw can change the inclination III, 318 denotes the connecting pin shaft IV, 319 denotes angle of a rectangular swing rod (308), which changes the walking leg, 320 denotes the body connecting pin shaft the motion locus of a stride length fork and controls II, 321 denotes the long slider, 322 denotes the connecting the horizontal movement of the leg end-point W pin shaft V, 323 denotes the connecting pin shaft VI, 324 denotes the short connecting rod, 325 denotes the (3) The four-bar linkage mechanism, defined by the path stride height fork, 326 denotes the sleeve, 327 denotes O ABO O in Figure 3, mainly provides motion con- 1 4 1 the body connection pin shaft IV, 328 denotes the motor nection and transmission between the stride length for the stride length adjusting, 329 denotes the body con- cam (306) and the stride height cam (315), which nection pin shaft V, 330 denotes the connection plate of transmits the motion of the driving motor for stride the motor for the stride length adjusting, 331 denotes length to the lifting mechanism the nut slider, and lastly, 332 denotes the connection pin (4) The lifting mechanism, defined by the path O O K shaft VII. 1 4 The structure of the single-leg walking system includes K O O in Figure 3, mainly determines the lifting 5 1 the following mechanisms (see Figures 2 and 3): distance of the leg end-point W. During the move- ment, a stride height cam (315) swings around the (1) The stride length mechanism, defined by the path robot body. Driven by the stride height cam, a stride O NN QRO O in Figure 3, mainly determines the height fork (325) carries out a swinging motion. This 1 6 1 motion causes the walking leg to lift through the long distance of the horizontal movement of the leg end- e 5 4 Applied Bionics and Biomechanics R1 x ' 1 휃 1 N r N' l1 휃 2 훼 r p rAB O3 O2 θ A 휃 6 훾 2 l7 훼 7 r6 O6 y4 O 7 n 훾 6 K 316 r4 R4 휃 4 r7 h' 훽 x 4 K' 휃 45 휑 4 O5 O4 x 4' r8 휃 8 휃 5 r5 휃 u 휃 p l 319 W2 Wi W3 Ground line W1 W4 Figure 3: The schematic of the single-leg walking mechanism. slider (321), controlling the lifting height of the moves one leg forward while the other three legs are station- motion path of the leg end-point W ary supporting the robot body and keeping the robot stable. When that leg completes the lifting, stretching forward, and (5) The walking mechanism, defined by the path O O S 1 5 dropping actions, the leg end-point W touches the ground. UTO O in Figure 3, represents a 2-DOF 5-bar link- 6 1 The motion path of the leg end-point W, shown in age mechanism. Input motions denote two swings of Figure 3, follows the trajectory from W to W , passing 1 4 the stride height fork (325) and the connecting rod through W and W , where W is the starting point of the 2 3 1 (316), and lifting and forward/backward motions leg end-point before lifting from the ground and W is the are formed, determining the motion path of the leg contact point of the leg end-point after touching the ground. end-point W After the leg completes its movement, it remains stable to support the robot body, while the other three legs perform the same movement in turn. After all four legs have com- 3. Working Principle of a Bionic Horse Robot pleted their movements and touched the ground, all four When a bionic horse robot walks, all its four legs need to driving motors for stride length on the four legs drive their move simultaneously. The lifting height of a leg is mainly respective legs at the same time. At this point, the end- points of the four legs remain fixed on the ground, while adjusted by the motor for the stride length adjusting. The driving motor for the stride length drives the stride length pushing the robot body to move forward or backward. The stride length cam rotates for one cycle, and the robot moves adjusting mechanism to produce the lifting and forward/- backward motions of the leg. forward or backward for one gait cycle. After one gait cycle The structure of the bionic horse robot is shown in is completed, each mechanism resets to its initial state and prepares for the next gait cycle. Figure 1, and when the bionic horse robot moves, it firstly 4 Applied Bionics and Biomechanics 5 W' W 50 W' W' W' 40 10 (b) (a) W' W 3 W 3 W' W W' 4 W' 4 1 W W' 6 60 W' 5 5 (c) (d) Figure 4: The gait of the bionic horse robot. Following Figures 2 and 3, and taking the leg forward ing legs perform the same movement one by one in a specific movement as an example, we have the following. The driving order, keeping the body relatively stable. motor for the stride length (303) drives the stride length cam When all the four legs have moved from point W to (306), and the motion of the stride length cam is divided into point W , then all of them are driven by the corresponding two motion loops. (1) The four-bar mechanism (O ABO O ) four driving motors for stride length simultaneously. At this 1 4 1 causes the stride height cam to swing around the robot body. point, the end-point of each leg W is fixed on the ground and Further, the motion of the stride height cam causes the stride pushes the robot body forward forming a robot body move- height fork to swing around the robot body. The motion of ment path (Figure 4(a)), where the mass center of the robot the stride height fork moves the long slider across the slide body moves along the trajectory W -W -W -W ; this 40 50 60 10 way of the leg. The motion of the long slider lifts the leg up trajectory reveals an antisymmetric relationship with the the- and mainly controls the lifting portion in the motion path oretical trajectory of the leg end-point W -W -W -W , 4 5 6 1 of the leg end-points. (2) The rotation of the stride length shown in Figure 4(c). In other words, while the leg is not cam causes the motion of the stride length fork, and the moving from the end-point W, a full cycle movement is com- motion of the stride length fork causes the connecting rod pleted by the antisymmetric motion of the mass center of the to swing around the robot body. The swinging of the con- robot body. After the robot body movement ends, the leg necting rod makes the leg move forward or backward. The end-point W of the previous cycle becomes the starting stride length fork is attached to a short slide block that moves point of the next cycle (new W ) and the robot body moves along the rectangular swing rod. Thus, by controlling the one step forward. rotation of the motor for the stride length adjusting, the incli- The trajectory of the leg end-point can be adjusted by the nation angle of the rectangular swing rod can be adjusted. motor for stride length adjusting. Generally, the motor for When the inclination angle of the rectangular swinging stride length adjusting is locked in the process of a movement rod changes, the movement of the short slide block will cause cycle, which means that it does not rotate. Namely, when the the connecting rod to swing around the robot body at a dif- motor for stride length adjusting rotates, angle α ,defined by ferent direction, which will result in the gait of the bionic the horizontal part of the rectangular swing rod and the hor- horse robot shown in Figure 4. In Figure 4, (a) shows the izontal plane (see Figure 3), changes due to the motion of the movement trajectory of the robot body in a forward gait, nut slider. Any variation in α will change the motion range (b) shows the movement trajectory of the robot body in a of the short slider on the rectangular swing rod; the motion backward gait, (c) shows the movement trajectory of the leg of the short slider will affect the motion of the stride length end-point in a forward gait, and (d) shows the movement tra- fork, so that the swing angle of the connecting rod will jectory of the leg end-point in a backward gait. change, altering the trajectory of the leg end-point and thus As can be seen in Figure 4, when the leg is driven by the adjusting the stride length of the robot. long slider and connecting rod, forming the leg end-point Usually, once the motor for stride length adjusting is set path from W to W , passing through W , W , W , and up, the trajectory of the leg end-point W is determined. 1 6 2 3 4 W , successively (Figure 4(c)), a forward gait is generated. The robot moves normally when the above movement is On the other hand, when the formed leg end-point path goes repeated continuously. ′ ′ ′ ′ ′ ′ from W to W , passing through W , W , W , and W 1 6 2 3 4 5 , successively (Figure 4(d)), a backward gait is generated. 4. Kinematic Modeling and Analysis of a Bionic Here, a forward gait is used as an example to explain the Horse Robot robot movement principle. As already mentioned, when the robot begins to move, three legs support the body, while A model for calculation of the trajectory of the leg end-point one leg moves following the trajectory defined by W , W , W is established based on the function of the stride length 1 2 W , and W , successively. Afterwards, the other three walk- cam and the stride height cam. 3 4 c 6 Applied Bionics and Biomechanics 4.1. Kinematic Relationships of a Constant-Breadth Three- Center Cam. Both the stride length cam and the stride height cam of the bionic horse robot use the constant-breadth three- center cam, whose structure is shown in Figure 5. In Figure 5, O'' it can be seen that the cam profile consists of six arc sections _ _ _ _ _ _ (HB, BC, CD, DF, FG, and GH). The centers of these arcs are ′ ″ ′ ″ 훽 O, O , O , O, O , and O , respectively. b A The motion parameters of the constant-breadth three- center cam are given in Table 1. In Table 1, a, b, and c denote the structural parameters of the constant-breadth three- O' ′ ″ center cam, O, O , and O are the rotation centers, N is the contact point between the cam and the follower, φ denotes the input angle, β denotes the output angle, and R denotes the radius vector. Figure 5: The constant-breadth three-center cam. In Table 1, it holds that where l is the initial length of the lead screw, θ is the rota- 0 3 tion angle of the motor for stride length adjusting, and α is −1 θ = 2 sin , 2a the angle between the lead screw and the horizontal direc- tion. Using equation (2), a relationship between α and θ ð1Þ 1 3 1, φ ≥ 0, can be obtained. SIGNφ = It should be noted that in equation (2), symbol “+” stands −1, φ <0: for the positive rotation of the motor for stride length adjust- ing and symbol “-” stands for the reverse rotation of the 4.2. Kinematic Analysis of a Stride Length Adjusting motor for stride length adjusting. Mechanism. The closed loop of the stride length adjusting mechanism is given as O O O O O (see Figure 3). The kine- 4.3. Kinematic Analysis of a Stride Length Mechanism. The 1 7 3 2 1 matic relations of the stride length adjusting mechanism can stride length mechanism is defined by O NN QRO O (see 1 6 1 be obtained by solving the following equation: Figure 3). Two closed loops, O NN QPO O and 1 7 1 O O PQO O , can be obtained by considering the stride 1 7 6 1 3 length adjusting mechanism. From the closed loop O NN X − X cos α + Y − Y sin α = l ± s , ð2Þ O O 1 O O 1 0 7 2 7 2 2π QPO O , we have 7 1 ! ! ! ! ! ! ð3Þ ′ ′ O N +NN +N Q =O O +O P + PQ , 1 1 7 7 j θ +φ +β +π/2 jðÞ θ +φ ðÞ jðÞ θ +φ +β +π jðÞ α +π/2 jα 1 1 1 10 1 1 1 1 1 10 ð4Þ R e + he + e e = X + Y j + r e + Se , 1 2 O O p 7 7 π π R cos θ + φ + h cos θ + φ +β + + e cos θ + φ +β + π = X + r cos α + + S cos α , ðÞ ðÞ 1 1 1 1 1 10 2 1 1 10 O p 1 1 2 2 ð5Þ π π R sinðÞ θ + φ + h sin θ + φ +β + + e sinðÞ θ + φ +β + π = Y + r sin α + + S sin α : 1 1 1 1 2 1 1 O p 1 1 1 10 10 2 2 Further, from the closed loop O O PQO O , we have where β = −β SIGNðφ Þ and θ = π/2 + θ + β + φ + γ , 1 7 6 1 10 1 1 2 1 10 1 2 and θ denotes the input angle of the driving motor for the ! ! ! ! ! ! ð6Þ stride length. When structural parameters a , b , and c of O O +O P + PQ + QR =O O +O R, 1 1 1 1 7 7 1 6 6 the stride length cam are determined, the values of φ , h, S, j α +π/2 jα θ jθ ðÞ and θ can be obtained by combining the relations given in 1 1 2 6 X + Y j + r e + Se + l e = X + Y j + r e , O O p 1 O O 6 7 7 6 6 Table 1 with equations (5) and (8) and using the Newton iter- ð7Þ ation method. X + r cos α + + S cos α + l cos θ = X + r cos θ , 4.4. Kinematic Analysis of a Four-Bar Linkage. From the O p 1 1 1 2 O 6 6 7 6 closed loop O ABO O (see Figure 3), we have 1 4 1 Y + r sin α + + S sin α + l sin θ = Y + r sin θ , O p 1 1 1 2 O 6 6 7 6 ! ! ! ! ð9Þ ð8Þ O A + AB =O O +O B, 1 1 4 4 Applied Bionics and Biomechanics 7 j θ +γ jθ j θ +γ ðÞ ðÞ where θ and θ can be obtained from equation (11) by using 1 1 A 4 4 A 4 r e + r e = X + Y j + r e , ð10Þ 1 AB O O 4 4 4 the Newton iteration method. r cosðÞ θ + γ + r cos θ = X + r cosðÞ θ + γ , 1 1 AB A O 4 4 1 4 4.5. Motion Analysis of a Lifting Mechanism. The closed loop r sinðÞ θ + γ + r sin θ = Y + r sinðÞ θ + γ , 1 1 AB A O 4 4 1 4 of the lifting mechanism is O O KK O O (see Figure 3), 1 4 5 1 ð11Þ and based on it, the following equations can be obtained: ! ! ! ! ! ð12Þ ′ ′ O O +O K +KK +K O =O O , 1 4 4 5 1 5 jðÞ θ +φ jðÞ θ +φ +β +π/2 jðÞ θ +φ +β +π 4 4 4 4 4 40 4 40 ′ ð13Þ X + Y j + R e + h e + e e = X + Y j, O O 4 5 O O 4 4 5 5 X + R cosðÞ θ + φ + h cos θ + φ + β + + e cosðÞ θ + φ + β + π = X , O 4 4 4 4 4 40 5 4 4 40 O 4 5 ð14Þ Y + R sinðÞ θ + φ + h sin θ + φ + β + + e sinðÞ θ + φ + β + π = Y , O 4 4 4 5 4 O 4 4 40 4 40 4 5 ′ 4.6. Motion Analysis of a Leg Mechanism. The closed loop of where β = −β SIGNðφ Þ, and φ and h can be obtained 40 4 4 4 the leg mechanism is O O SUTO O (see Figure 3). From from equation (14) by using the Newton iteration method. 1 5 6 1 this closed loop, we have ! ! ! ! ! ! ð15Þ O O + O S =O O +O T + TU + US , 1 5 5 1 6 6 jθ j θ +γ jθ j θ +θ −π ðÞ ðÞ 5 6 8 U 8 ð16Þ X + Y j + r e = X + Y j + r e + r e + le , O O 5 O O 7 8 5 5 6 6 X + r cos θ = X + r cos θ + γ + r cos θ + l cos θ + θ − π , ðÞ ðÞ O 5 5 O 7 6 6 8 8 U 8 5 6 ð17Þ Y + r sin θ = Y + r sin θ + γ + r sin θ + l sin θ + θ − π , ðÞ ðÞ O 5 5 O 7 6 6 8 8 U 8 5 6 where θ =3π/2 + θ + β + φ − γ , and θ and l can be 5. Analysis and Discussion of Pressure 5 4 8 40 4 5 obtained from equation (17) also by using the Newton itera- Angle Performance tion method. The pressure angle refers to a sharp angle formed between the normal on the cam profile on the contact point and the 4.7. Trajectory Coordinate of Leg End-Point. Following the velocity direction of the corresponding contact point of the schematic of the single-leg walking mechanism presented in follower. In the following, the pressure angle of the stride Figure 3, and solving equations (5), (8), (11), (14), and (17), length cam in the cam-linkage mechanism is discussed. the parameters l , α , θ , h, S, θ , and θ can be calculated, 7 1 6 5 8 The calculation model of the pressure angle of the stride and the trajectory coordinates of the leg end-point can be length cam is illustrated in Figure 6. According to the obtained by three-center theorem of the velocity instantaneous center, the three velocity instantaneous centers of three adjacent components must be on the same straight line. If the velocity X = X + r cosðÞ θ + γ + r cos θ + S cosðÞ π + θ − θ , W O 7 6 6 8 8 8 8 P instantaneous centers of two adjacent component pairs are Y = Y + r sinðÞ θ + γ + r sin θ + S sinðÞ π + θ − θ , determined, the velocity instantaneous center of another pair W O 7 6 6 8 8 8 8 P of components can be obtained based on the mentioned the- ð18Þ orem. It can be determined that point P is the velocity instantaneous center of the stride length fork that touches the stride length cam (Figure 6). where X and Y are the trajectory coordinates of the leg W W The pressure angle α can be calculated by setting up the end-point W in the horizontal and vertical directions, respectively. loop equations. From the closed loop O PQP O O , we have 7 6 7 8 Applied Bionics and Biomechanics ! ! ! ! ! ð19Þ ′ ′ O O +O P =O P + PQ +QP , 7 6 6 7 jθ jðÞ α +π/2 jðÞ α +π jðÞ α +3π/2 6 1 1 1 ð20Þ X − X + Y − Y j + l e = r e + Se + l e , O O O O 4 P 3 6 7 6 7 π 3π X − X + l cos θ = r cos α + + S cos α + π + l cos α + , ðÞ O O 4 6 P 1 1 3 1 < 6 7 2 2 ð21Þ π 3π : Y − Y + l sin θ = r sin α + + S sinðÞ α + π + l sin α + : O O 4 6 P 1 1 3 1 6 7 2 2 the rectangular swing rod, and the obtained results are pre- From the closed loop O O P NO , we have 1 6 1 sented in Figures 7–9. ! ! As can be seen in Figure 9, when α increased, the height ! ! ð22Þ ′ ′ O O +O P +P N =O N, 1 6 6 1 of the leg end-point trajectory changed significantly. ° ° ° The inclination angles of 12.5 , 14.5 , and 18.5 were used jθ jθ jðÞ θ +φ 6 N 1 ð23Þ X + Y j + l e + l e = R e , to analyze and explain the changes in the leg end-point tra- O O 4 2 1 6 6 jectory and the body trajectory, and analysis results are pre- X + l cos θ + l cos θ = R cos θ + φ , ðÞ sented in Figures 10–12. O 4 6 2 N 1 1 1 ð24Þ As can be seen in Figure 10, at the inclination angle of Y + l sin θ + l sin θ = R sin θ + φ : ðÞ ° O 4 6 2 N 1 1 1 12.5 , the maximum lifting height of the leg end-point W was 47.72 mm and the maximum horizontal moving distance By solving equations (21) and (24), l , l , l , and θ can be 2 3 4 N was 506.33 mm. For the robot body, the maximum change in obtained. Then, it holds that height was 67.76 mm, the maximum horizontal moving dis- tance was 631.22 mm, and the stride length of this kind of α = θ + −ðÞ θ + φ + β , ð25Þ N 1 1 10 gait was 417.18 mm. The results obtained at the inclination angle of 14.5 are presented in Figure 11, where it can be seen that the where β = −β SIGNðφ Þ. 10 1 1 maximum lifting height of the leg end-point W was 139.68 mm and the maximum horizontal moving distance 6. Calculation Example and Simulation was 611.79 mm. For the robot body, the maximum change in height was 67.13 mm, the maximum horizontal moving In this section, a calculation example and a motion simula- tion are given. The structural parameters of the designed distance was 555.32 mm, and the stride length of this kind single-leg system were as follows. Note that all units are of gait was 548.59 mm. Lastly, the results obtained at the inclination angle of 18.5 given in mm. The center O of the stride length cam was used as the origin of the coordinate system (see Figure 3), are shown in Figure 12, where it can be seen that the maxi- mum lifting height of W was 1293.14 mm and the maximum and the coordinates of the points were as follows: O (-360, -753), O (-460, -720), O (383, -531), O (-114, -654), and horizontal moving distance was 852.04 mm. For the robot 2 6 4 body, the maximum change in height was 30.62 mm, the max- O (-433, -923). The lengths where as follows: r = 185 5 p imum horizontal moving distance was 232.85 mm, and the (length of O P), l =240 (length of QR), e =165 (single-side 7 1 2 stride length of this kind of gait was 226.83 mm. Based on width of the stride length fork), r =44 (length of BO ), r = 4 4 1 the obtained results it can be concluded that when this kind 35 (length of O A), r =655 (length of AB), r =474 (length 1 AB 7 of leg end-point trajectory (shown in Figure 12) is used for of O T), r =95 (length of RO ), r = 100 (length of TU), 6 6 6 8 the gait of the bionic horse robot, a large lifting height of the S = 1081 (distance between point U and point W), r = 356 8 5 leg can be achieved, but the moving distance of the body will (length of O S), and e = 120 (single-side width of the stride 5 5 be small, which is similar to the horse strolling in situ. height fork). Additionally, the angles were γ = 180 , γ = 1 2 ° ° ° ° ° ° Generally speaking, at α =14:5 , this leg end-point tra- 40 , γ =62 , γ =40 , γ = −8 , θ = 120 , and θ = 125 . 1 5 6 4 U P jectory (shown in Figure 11) for the gait of the bionic horse The structural parameters of the stride cam were a = 220, robot would be a better choice. b = 160, and c =25, while those of the stride height cam 1 1 The pressure angles of the stride length cam mechanism were a = 220, b = 220, and c =10. MATLAB software was 2 2 2 during the forward gait were also determined, and the used for the development of the motion analysis system for obtained values are presented graphically in Figure 13. In the bionic horse. Figure 13, the rotation angle of the stride length cam is shown 6.1. Calculation Example. Based on the kinematic model of on the horizontal axis and the inclination angle is presented on the vertical axis; different line types are used to show the the bionic horse robot and the related input parameters, the change in the motion trajectory and the pressure angle for pressure angle for different values of the inclination angle. one cycle was calculated. Several trajectories of the leg end- In general, the maximum allowed pressure angle was 45 . point were obtained by changing the inclination angle α of Also, as can be seen in Figure 13, the maximum angle of the 1 r Applied Bionics and Biomechanics 9 Table 1: Motion parameters of the constant-breadth three-center cam. Absolute value of input anglejj φ Output angle β Radius vector R −1 jj φ ≤ sin 0 2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a − b /4 a θ a ⋅ sinðÞ φSIGNφ − β −ðÞ θ /2 −1 x x b π −1 −1 sin ⋅ sin φSIGNφ − sin < φ ≤ + tan jj a + c 2 sin β 2a 2 a + c − b/2 ðÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a θ a − b /4 a ⋅ sinðÞ φSIGNφ − β +ðÞ θ /2 −1 x x π b −1 −1 sin ⋅ sin φSIGNφ + + tan < φ ≤ π‐sin jj a + c‐b 2 sin β 2 a + c − b/2 2a ðÞ −1 2a + c − b π − sin <jj φ ≤ π 0 2a −700 R −800 ' −900 −1000 −1100 1 1 훼 −1200 −1300 −1400 −1500 N' −1600 −1700 −1800 −1900 S −2000 −2100 −2200 휃 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 휃 (°) P' 훼 = 12.5° 훼 = 16.5° 1 1 훼 = 13.5° 훼 = 17.5° 훼 = 14.5° 훼 = 18.5° 1 1 훼 = 15.5° Figure 6: Calculation model of the pressure angle of the stride Figure 8: The vertical direction trajectories of the leg end-point for length cam. moving forward at different inclination angles. −800 −900 −1000 −1100 −1200 −1300 −1400 −1500 −1600 −1700 −1800 −1900 −2000 −2100 −2200 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 200 300 400 500 600 700 800 900 1000 휃 (°) X (mm) 훼 = 12.5° 훼 = 16.5° 훼 = 12.5° 훼 = 16.5° 1 1 1 1 훼 = 13.5° 훼 = 17.5° 훼 = 13.5° 훼 = 17.5° 1 1 1 1 훼 = 14.5° 훼 = 18.5° 훼 = 14.5° 훼 = 18.5° 1 1 1 1 훼 = 15.5° 훼 = 15.5° Ground line 1 1 Figure 7: The horizontal direction trajectories of the leg end-point Figure 9: The trajectories of the leg end-point for moving forward for moving forward at different inclination angles. at different inclination angles. X (mm) Y (mm) Y (mm) w w 10 Applied Bionics and Biomechanics −720 −700 40 W −740 −800 W 60 −760 200 300 400 500 600 700 800 900 1000 1100 X (mm) −780 w W −800 −800 −900 200 300 400 500 600 700 800 900 1000 1100 −1000 −1100 X (mm) −1200 −2020 −1300 −2040 −1400 2 −1500 −2060 −1600 −2080 −1700 1 W 4 2 −1800 Ground line −2100 Ground line 3 W −1900 −2120 W −2000 1 W 6 4 −2100 W W −2140 5 −2200 200 300 400 500 600 700 800 900 1000 1100 200 300 400 500 600 700 800 900 1000 1100 X (mm) X (mm) Figure 12: The trajectories of the leg end-point (bottom) and the Figure 10: The trajectories of the leg end-point (bottom) and the body (top) at α =18:5 . body (top) at α =12:5 . −740 −760 −780 10 22 −800 200 300 400 500 600 700 800 900 1000 1100 X (mm) −1920 −1940 W 17 −1960 −1980 −2000 −2020 −2040 −2060 −2080 −2100 10 W 5 6 Ground line −2120 9 −2140 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 휃 (°) 200 300 400 500 600 700 800 900 1000 1100 X (mm) 훼 = 12.5° 훼 = 16.5° 1 1 훼 = 13.5° 훼 = 17.5° 1 1 Figure 11: The trajectories of the leg end-point (bottom) and the 훼 = 14.5° 훼 = 18.5° 1 1 body (top) at α =14:5 . 훼 = 15.5° Figure 13: Pressure angles of the stride length cam mechanism stride length cam mechanism was less than 26 , which was during the forward gait for different inclination angles. within the allowed range. The forward motion simulation for one leg is shown in Figure 15. The motion trajectory of the leg and the pressure 6.2. Motion Simulation. In order to validate the calculated values of the motion trajectory and pressure angle of the angle of the cam were automatically exported by the simula- bionic horse, a structural model of the robot was developed, tion software. In Figure 16, the trajectory of the leg end-point W during and its motion was simulated in Solidworks software. The ° ° model of the forward leg motion at α =14:5 of the bionic the forward gait at α =14:5 obtained by both the motion simulation (solid line) and the theoretical calculation (dotted horse robot in the simulation software is presented in Figures 14 and 15. line) is shown. As can be seen in Figure 16, the maximum dif- A forward motion cycle simulation for the robot entirety ferences between the motion simulation trajectory and theo- is shown in Figure 14, (a) shows the initial position of the retically calculated trajectory were 4.60 mm in the x-direction robot, (b) shows the forward movement of the first leg, (c) and 2.15 mm in the y-direction. In Figure 17, the pressure angle curve of the stride length shows the forward movement of the second leg, (d) shows the forward movement of the third leg, (e) shows the forward cam during the forward gait at α =14:5 obtained by both movement of the fourth leg, and (f) shows the movement of the motion simulation (solid line) and the theoretical calcula- the robot body completing a forward gait. tion (dotted line) is shown. The maximum differences in the Y (mm) Y (mm) Y (mm) Y (mm) w w w w (°) Y (mm) Y (mm) w w Applied Bionics and Biomechanics 11 (a) (b) (c) (d) (e) (f) Figure 14: A forward motion cycle simulation for the robot entirety at α =14:5 in Solidworks software. Figure 15: The forward leg motion simulation at α =14:5 in Solidworks software. −1920 −1940 −1960 −1980 W 15 −2000 −2020 −2040 W 13 −2060 W 4 −2080 −2100 W Ground line −2120 −2140 200 300 400 500 600 700 800 900 1000 1100 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 X (mm) 휃 (°) Theoretical value Theoretical value Actual value Actual value Figure 16: The trajectory of the leg end-point during the forward Figure 17: The pressure angle of the stride length cam during the ° ° gait at α =14:5 . forward gait at α =14:5 . 1 1 Y (mm) (°) 12 Applied Bionics and Biomechanics pressure angle between the motion simulation and the theo- IEEE International Conference on Robotics and Automation, pp. 4736–4741, Anchorage, AK, USA, 2010. retical calculation were 0.21 . Based on the results demonstrated in Figures 16 and [3] W. Guobiao, C. Diansheng, C. Kewei, and Z. Ziqiang, “The current research status and development strategy on bio- 17, the differences in results between the motion simula- mimetic robot,” Journal of Mechanical Engineering, vol. 51, tion and the theoretical calculation were minimal; thus, the no. 13, pp. 27–44, 2015. motion trajectory and pressure angle calculation models [4] J. Buchli, J. E. Pratt, and N. Roy, “Editorial: Special issue on were verified. legged locomotion,” International Journal of Robotics Research, vol. 30, no. 2, pp. 139-140, 2011. 7. Conclusion [5] L. Mantian, J. Zhenyu, G. Wei, and S. Lining, “Leg prototype of a bio-inspired quadruped robot,” Robot, vol. 36, no. 1, pp. 21– In this study, a quadruped bionic horse robot driven by a 28, 2014. cam-linkage mechanism is proposed. It is demonstrated that [6] S. Chen, X. Chen, Z. Liu, and X. 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Bohannon, “Treatment interventions for the paretic upper limb of stroke survivors: a Data Availability critical review,” Neurorehabilitation and Neural Repair, vol. 17, no. 4, pp. 220–226, 2003. The data used to support the findings of this study are [13] S. Viteckova, P. Kutilek, and M. Jirina, “Wearable lower limb available from the corresponding author upon request. robotics: a review,” Biocybernetics and Biomedical Engineering, vol. 33, no. 2, pp. 96–105, 2013. Conflicts of Interest [14] P. Y. Cheng and P. Y. Lai, “Comparison of exoskeleton robots and end-effector robots on training methods and gait biome- The authors declare that they have no conflicts of interest. chanics,” in Intelligent Robotics and Applications. ICIRA 2013, J. Lee, M. C. Lee, H. Liu, and J. H. Ryu, Eds., vol. 8102 Acknowledgments of Lecture Notes in Computer Science, pp. 258–266, Springer, Berlin, Heidelberg, 2013. The research presented in this work was supported by the [15] S. Fisher, L. Lucas, and T. A. Thrasher, “Robot-assisted Key Science and Technology Research Project of the Henan gait training for patients with hemiparesis due to stroke,” Province (182102210159), China, and the Doctoral Research Topics in Stroke Rehabilitation, vol. 8, no. 3, pp. 269–276, Funded Projects of the Zhengzhou University of Light Indus- try (2016BSJJ009). [16] P. Alves, F. Carballocruz, L. F. Silva, and P. Flores, “Synthesis of a mechanism for human gait rehabilitation: an introductory approach,” in New Trends in Mechanism and Machine Science, References vol. 24 of Mechanisms and Machine Science, pp. 121–128, [1] M. Raibert, K. Blankespoor, G. Nelson, and R. Playter, “Big- Springer, Cham, 2015. Dog, the rough-terrain quadruped robot,” IFAC Proceedings [17] B. Y. Tsuge, M. M. Plecnik, and J. Michael Mccarthy, “Homo- Volumes, vol. 41, no. 2, pp. 10822–10825, 2008. topy directed optimization to design a six-bar linkage for a [2] D. Wooden, M. Malchano, K. Blankespoor, A. 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Conceptual Design and Computational Modeling Analysis of a Single-Leg System of a Quadruped Bionic Horse Robot Driven by a Cam-Linkage Mechanism

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Copyright © 2019 Liangwen 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.
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10.1155/2019/2161038
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Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 2161038, 13 pages https://doi.org/10.1155/2019/2161038 Research Article Conceptual Design and Computational Modeling Analysis of a Single-Leg System of a Quadruped Bionic Horse Robot Driven by a Cam-Linkage Mechanism Liangwen Wang , Weiwei Zhang, Caidong Wang , Fannian Meng , Wenliao Du , and Tuanhui Wang School of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou 450002, China Correspondence should be addressed to Liangwen Wang; w_liangwen@sina.com and Caidong Wang; vwangcaidong@163.com Received 24 May 2019; Accepted 3 October 2019; Published 4 November 2019 Academic Editor: Alberto Borboni Copyright © 2019 Liangwen 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. In this study, the configuration of a bionic horse robot for equine-assisted therapy is presented. A single-leg system with two degrees of freedom (DOFs) is driven by a cam-linkage mechanism, and it can adjust the span and height of the leg end-point trajectory. After a brief introduction on the quadruped bionic horse robot, the structure and working principle of a single-leg system are discussed in detail. Kinematic analysis of a single-leg system is conducted, and the relationships between the structural parameters and leg trajectory are obtained. On this basis, the pressure angle characteristics of the cam-linkage mechanism are studied, and the leg end-point trajectories of the robot are obtained for several inclination angles controlled by the rotation of the motor for the stride length adjusting. The closed-loop vector method is used for the kinematic analysis, and the motion analysis system is developed in MATLAB software. The motion analysis results are verified by a three-dimensional simulation model developed in Solidworks software. The presented research on the configuration, kinematic modeling, and pressure angle characteristics of the bionic horse robot lays the foundation for subsequent research on the practical application of the proposed bionic horse robot. 1. Introduction researchers have extensively studied the structure, move- ment, and control of the robot legs. Li et al. [5] systemati- The quadruped walking robots are an important type of cally studied a single-leg system of a quadruped bionic robot. Based on the analysis of the muscle-bone structure legged robots. Recently, many countries have conducted profound research on the walking robots, which denote a of quadrupeds, the DOF configuration of a single leg was frontier technology of the strategic significance. The BigDog determined. Chen et al. [6] studied a bionic quadruped robot developed by Boston Dynamics is a rough-terrain robot that in order to improve its dynamic stability and adaptability by captures the mobility, autonomy, and speed of living crea- imitating the quadrupeds. Chen et al. [7] designed a new tures, which is a typical example of a legged robot [1, 2]. bionic robot named Hound. The body structure, especially The cheetah robots presented in [3] are capable of many that of the legs, and geometric relationships of the Hound functions, among which are sprinting and sharp turning, robot were designed based on the bionic research. Anantha- which is similar to the kinematics of the biological proto- narayanan et al. [8] designed the bionic legs suitable for types. The quadruped walking robots are equipped with high-speed motion, taking into consideration the balance multibranched motion mechanisms. The multi-degree-of- between weight and strength. Smith and Jivraj [9] compared freedom coupling between the branches makes the coordina- a hind leg with a leg with a different arrangement, demon- tion of robot motion very complicated [4]. In order to strating in detail how leg orientation can affect the dynamic improve the mobility and load capacity of a robot, several characteristics and gait performance of a robot. Seok et al. 2 Applied Bionics and Biomechanics tions for the double-dwell follower were used, basically syn- [10] introduced the design principles of the highly efficient walking robots, which were implemented in the design and thesized applying the cycloidal and polynomial motion experimental analysis of a cheetah robot. curves to a monomial basis [26]. Rothbart [27] presented a constant-breadth circular-arc cam profile in which the fol- Nowadays, it is relatively common to use robots in human rehabilitation and treatment processes. The robotized lower movement denoted a double-dwell function. Qian systems have been used for rehabilitation, improving the [28] investigated a constant-diameter cam mechanism, which efficacy while reducing the healthcare costs [11]. The reha- included a double roller follower with a planar movement that bilitation robots have significant advantages in helping was joined to an output rocker arm, and used a graphical- analytical method to establish the relationship between the patients to recover from neurological disorders and to become able to walk again [12]. Over the last few decades, geometric parameters of the designed constant-diameter many kinds of lower limb rehabilitation robots have been cam mechanism and the cam angles. developed [13–15]; however, their high cost makes them The circular-arc cams are easy to design, manufacture, impracticable for home healthcare. and test, which makes them cheaper than others [29, 30]. Cardona et al. [31] studied two constant-breadth cam mech- Recently, several one-DOF mechanisms have been pro- posed for a rehabilitation process, such as the four-bar anisms and presented the equations for calculating the cam mechanisms proposed by Alves et al. [16], the Stephenson breadth when the translating follower is eccentric with an III six-bar mechanism [17], and the ten-bar linkage mecha- inclination, the radius of curvature of its profile and the slid- nism proposed by Tsuge and Mccarthy [18]. With the aim ing velocities of constant-breadth cam mechanisms with translating and oscillating followers. to overcome the limitation of such mechanisms that the desired gait path can be matched only approximately, Kay In the proposed constant-breadth three-center cam and Haws [19] proposed a cam-linkage mechanism, combin- design, a swing center point of the follower can slide along ing a fixed cam and a four-bar linkage to generate the desired the guide way, resulting in a complex plane follower motion. path accurately. Mundo et al. [20] synthesized the cam- Recently, there have been a few reports on the cam-linkage mechanism where a constant-breadth three-center cam was linkage mechanisms with one or more cams for precise path generation, while Soong [21] proposed a novel cam-geared used [31]. mechanism for path generation. Shao et al. [22] designed From the aspects of the overall robot structure, the struc- ture and the working principle of a single-leg system of the a new robot structure, where a seven-bar crank-slider mech- anism was combined with a cam to generate a precise target robot driven by a cam-linkage mechanism are discussed in detail. The kinematics modeling and analysis of the cam- path. The novel lower-limb rehabilitation system was com- posed of a body weight support system to unload the body linkage mechanism are carried out, and the relationships weight and two cam-linkage mechanisms to generate the between the structural parameters and foot trajectory of a single leg are obtained. On this basis, the pressure angle char- natural gait trajectory and guide the feet of a patient. As an unconventional therapy, an equine-assisted ther- acteristics of the cam-linkage mechanism are thoroughly investigated. The closed-loop vector method is used for the apy provides horse riding training [23], which allows the user’s pelvis and torso to feel the movement of the horse in kinematic analysis, and MATLAB software is used for the order to improve balance control, promote trunk extension, development of the motion analysis system. Moreover, the trajectory cluster of a robot meeting the pressure angle condi- make a rhythmic rotation of the trunk, and enhance endur- ance performance and cardiopulmonary function [24, 25]. tions is obtained. The motion analysis results are verified by the simulation with a three-dimensional model established At present, it is costly to use horses for equine-assisted ther- apy, so equestrian therapy is used as a relatively less-costly in Solidworks software. solution. In this study, in order to imitate the movement of The rest of the paper is organized as follows. In Section 2, the structure of the bionic horse robot driven by a cam- a horse and achieve effective equine-assisted therapy, a bionic horse robot driven by a cam-linkage mechanism is designed. linkage mechanism is briefly introduced. The working prin- Each leg of the proposed bionic horse robot has two driv- ciple of the proposed robot is explained in detail in Section 3. ing motors, which can adjust the span and height of the leg In Section 4, the proposed cam-linkage mechanism is pre- motion path. In the normal walking by using the leg motion, sented and analyzed. In Section 5, the pressure angle charac- teristics of the proposed robot are studied. The calculation only one driving motor is needed. The designed cam-linkage mechanism uses a constant- example, the motion simulation, and the obtained results breadth three-center cam. The constant-breadth cam mech- are given in Section 6. Finally, the conclusions are drawn anism belongs to the class of desmodromic or positive drive in Section 7. mechanisms. In the case of a parallel flat-faced double follower, the distance of contact points between the two 2. Structure of a Bionic Horse Robot Driven by a flat-faced followers and cam denotes the cam breadth. The Cam-Linkage Mechanism constant-breadth cams may be the circular-arc cams or hav- ing an arbitrary geometry and can operate either translating Following the bionic theory, a bionic horse robot imitates or oscillating a follower if the appropriate desmodromic con- the movement of a horse. A single-leg motion of a bionic ditions are established. horse robot is driven by a cam-linkage mechanism. The In the studies on the constant-breadth cam mechanisms, overall structure of the robot is presented in Figure 1, usually, the circular-arc cam profiles and displacement func- where it can be seen that the robot consists of the robot Applied Bionics and Biomechanics 3 301 302 303 304 305 307 308 323 311 324 312 326 314 327 315 328 316 329 317 330 318 331 319 1 2 3 332 320 Figure 1: The structure of the bionic horse robot. body (denoted by 1), the control system (denoted by 2), and the single-leg walking system (denoted by 3). The developed robot had four parallel and symmetrical single-leg walking Figure 2: The structure of the single-leg walking system. systems placed on both sides of the robot body. The structure of the single-leg walking system and the schematic of the single-leg walking mechanism are shown in Figures 2 and point W. During walking, driven by a driving motor 3, respectively. for the stride length (303), astride length fork (305) In Figures 2 and 3, 301 denotes the connection plate, 302 moves and swings forward or backward along with denotes the connection bolt, 303 denotes the driving motor the rotation of astride length cam. The motion of for the stride length, 304 denotes the drive shaft of the stride the stride length fork makes a connecting rod (316) length cam, 305 denotes the stride length fork, 306 denotes to rotate, thus moving the walking leg (319) forward the stride length cam, 307 denotes the fork connecting rod, or backward 308 denotes the rectangular swing rod, 309 denotes the short (2) The stride length adjusting mechanism, defined by slide block, 310 denotes the connecting pin shaft I, 311 the path O O O O O in Figure 3, mainly controls 1 7 3 2 1 denotes the lead screw, 312 denotes the coupler, 313 denotes the horizontal movement of the leg end-point W.A the body connecting pin shaft I, 314 denotes the connecting motor for the stride length adjusting (328) is con- pin shaft II, 315 denotes the stride height cam, 316 denotes nected with a lead screw (311) through a coupler. the connecting rod, 317 denotes the connecting pin shaft Rotation of the lead screw can change the inclination III, 318 denotes the connecting pin shaft IV, 319 denotes angle of a rectangular swing rod (308), which changes the walking leg, 320 denotes the body connecting pin shaft the motion locus of a stride length fork and controls II, 321 denotes the long slider, 322 denotes the connecting the horizontal movement of the leg end-point W pin shaft V, 323 denotes the connecting pin shaft VI, 324 denotes the short connecting rod, 325 denotes the (3) The four-bar linkage mechanism, defined by the path stride height fork, 326 denotes the sleeve, 327 denotes O ABO O in Figure 3, mainly provides motion con- 1 4 1 the body connection pin shaft IV, 328 denotes the motor nection and transmission between the stride length for the stride length adjusting, 329 denotes the body con- cam (306) and the stride height cam (315), which nection pin shaft V, 330 denotes the connection plate of transmits the motion of the driving motor for stride the motor for the stride length adjusting, 331 denotes length to the lifting mechanism the nut slider, and lastly, 332 denotes the connection pin (4) The lifting mechanism, defined by the path O O K shaft VII. 1 4 The structure of the single-leg walking system includes K O O in Figure 3, mainly determines the lifting 5 1 the following mechanisms (see Figures 2 and 3): distance of the leg end-point W. During the move- ment, a stride height cam (315) swings around the (1) The stride length mechanism, defined by the path robot body. Driven by the stride height cam, a stride O NN QRO O in Figure 3, mainly determines the height fork (325) carries out a swinging motion. This 1 6 1 motion causes the walking leg to lift through the long distance of the horizontal movement of the leg end- e 5 4 Applied Bionics and Biomechanics R1 x ' 1 휃 1 N r N' l1 휃 2 훼 r p rAB O3 O2 θ A 휃 6 훾 2 l7 훼 7 r6 O6 y4 O 7 n 훾 6 K 316 r4 R4 휃 4 r7 h' 훽 x 4 K' 휃 45 휑 4 O5 O4 x 4' r8 휃 8 휃 5 r5 휃 u 휃 p l 319 W2 Wi W3 Ground line W1 W4 Figure 3: The schematic of the single-leg walking mechanism. slider (321), controlling the lifting height of the moves one leg forward while the other three legs are station- motion path of the leg end-point W ary supporting the robot body and keeping the robot stable. When that leg completes the lifting, stretching forward, and (5) The walking mechanism, defined by the path O O S 1 5 dropping actions, the leg end-point W touches the ground. UTO O in Figure 3, represents a 2-DOF 5-bar link- 6 1 The motion path of the leg end-point W, shown in age mechanism. Input motions denote two swings of Figure 3, follows the trajectory from W to W , passing 1 4 the stride height fork (325) and the connecting rod through W and W , where W is the starting point of the 2 3 1 (316), and lifting and forward/backward motions leg end-point before lifting from the ground and W is the are formed, determining the motion path of the leg contact point of the leg end-point after touching the ground. end-point W After the leg completes its movement, it remains stable to support the robot body, while the other three legs perform the same movement in turn. After all four legs have com- 3. Working Principle of a Bionic Horse Robot pleted their movements and touched the ground, all four When a bionic horse robot walks, all its four legs need to driving motors for stride length on the four legs drive their move simultaneously. The lifting height of a leg is mainly respective legs at the same time. At this point, the end- points of the four legs remain fixed on the ground, while adjusted by the motor for the stride length adjusting. The driving motor for the stride length drives the stride length pushing the robot body to move forward or backward. The stride length cam rotates for one cycle, and the robot moves adjusting mechanism to produce the lifting and forward/- backward motions of the leg. forward or backward for one gait cycle. After one gait cycle The structure of the bionic horse robot is shown in is completed, each mechanism resets to its initial state and prepares for the next gait cycle. Figure 1, and when the bionic horse robot moves, it firstly 4 Applied Bionics and Biomechanics 5 W' W 50 W' W' W' 40 10 (b) (a) W' W 3 W 3 W' W W' 4 W' 4 1 W W' 6 60 W' 5 5 (c) (d) Figure 4: The gait of the bionic horse robot. Following Figures 2 and 3, and taking the leg forward ing legs perform the same movement one by one in a specific movement as an example, we have the following. The driving order, keeping the body relatively stable. motor for the stride length (303) drives the stride length cam When all the four legs have moved from point W to (306), and the motion of the stride length cam is divided into point W , then all of them are driven by the corresponding two motion loops. (1) The four-bar mechanism (O ABO O ) four driving motors for stride length simultaneously. At this 1 4 1 causes the stride height cam to swing around the robot body. point, the end-point of each leg W is fixed on the ground and Further, the motion of the stride height cam causes the stride pushes the robot body forward forming a robot body move- height fork to swing around the robot body. The motion of ment path (Figure 4(a)), where the mass center of the robot the stride height fork moves the long slider across the slide body moves along the trajectory W -W -W -W ; this 40 50 60 10 way of the leg. The motion of the long slider lifts the leg up trajectory reveals an antisymmetric relationship with the the- and mainly controls the lifting portion in the motion path oretical trajectory of the leg end-point W -W -W -W , 4 5 6 1 of the leg end-points. (2) The rotation of the stride length shown in Figure 4(c). In other words, while the leg is not cam causes the motion of the stride length fork, and the moving from the end-point W, a full cycle movement is com- motion of the stride length fork causes the connecting rod pleted by the antisymmetric motion of the mass center of the to swing around the robot body. The swinging of the con- robot body. After the robot body movement ends, the leg necting rod makes the leg move forward or backward. The end-point W of the previous cycle becomes the starting stride length fork is attached to a short slide block that moves point of the next cycle (new W ) and the robot body moves along the rectangular swing rod. Thus, by controlling the one step forward. rotation of the motor for the stride length adjusting, the incli- The trajectory of the leg end-point can be adjusted by the nation angle of the rectangular swing rod can be adjusted. motor for stride length adjusting. Generally, the motor for When the inclination angle of the rectangular swinging stride length adjusting is locked in the process of a movement rod changes, the movement of the short slide block will cause cycle, which means that it does not rotate. Namely, when the the connecting rod to swing around the robot body at a dif- motor for stride length adjusting rotates, angle α ,defined by ferent direction, which will result in the gait of the bionic the horizontal part of the rectangular swing rod and the hor- horse robot shown in Figure 4. In Figure 4, (a) shows the izontal plane (see Figure 3), changes due to the motion of the movement trajectory of the robot body in a forward gait, nut slider. Any variation in α will change the motion range (b) shows the movement trajectory of the robot body in a of the short slider on the rectangular swing rod; the motion backward gait, (c) shows the movement trajectory of the leg of the short slider will affect the motion of the stride length end-point in a forward gait, and (d) shows the movement tra- fork, so that the swing angle of the connecting rod will jectory of the leg end-point in a backward gait. change, altering the trajectory of the leg end-point and thus As can be seen in Figure 4, when the leg is driven by the adjusting the stride length of the robot. long slider and connecting rod, forming the leg end-point Usually, once the motor for stride length adjusting is set path from W to W , passing through W , W , W , and up, the trajectory of the leg end-point W is determined. 1 6 2 3 4 W , successively (Figure 4(c)), a forward gait is generated. The robot moves normally when the above movement is On the other hand, when the formed leg end-point path goes repeated continuously. ′ ′ ′ ′ ′ ′ from W to W , passing through W , W , W , and W 1 6 2 3 4 5 , successively (Figure 4(d)), a backward gait is generated. 4. Kinematic Modeling and Analysis of a Bionic Here, a forward gait is used as an example to explain the Horse Robot robot movement principle. As already mentioned, when the robot begins to move, three legs support the body, while A model for calculation of the trajectory of the leg end-point one leg moves following the trajectory defined by W , W , W is established based on the function of the stride length 1 2 W , and W , successively. Afterwards, the other three walk- cam and the stride height cam. 3 4 c 6 Applied Bionics and Biomechanics 4.1. Kinematic Relationships of a Constant-Breadth Three- Center Cam. Both the stride length cam and the stride height cam of the bionic horse robot use the constant-breadth three- center cam, whose structure is shown in Figure 5. In Figure 5, O'' it can be seen that the cam profile consists of six arc sections _ _ _ _ _ _ (HB, BC, CD, DF, FG, and GH). The centers of these arcs are ′ ″ ′ ″ 훽 O, O , O , O, O , and O , respectively. b A The motion parameters of the constant-breadth three- center cam are given in Table 1. In Table 1, a, b, and c denote the structural parameters of the constant-breadth three- O' ′ ″ center cam, O, O , and O are the rotation centers, N is the contact point between the cam and the follower, φ denotes the input angle, β denotes the output angle, and R denotes the radius vector. Figure 5: The constant-breadth three-center cam. In Table 1, it holds that where l is the initial length of the lead screw, θ is the rota- 0 3 tion angle of the motor for stride length adjusting, and α is −1 θ = 2 sin , 2a the angle between the lead screw and the horizontal direc- tion. Using equation (2), a relationship between α and θ ð1Þ 1 3 1, φ ≥ 0, can be obtained. SIGNφ = It should be noted that in equation (2), symbol “+” stands −1, φ <0: for the positive rotation of the motor for stride length adjust- ing and symbol “-” stands for the reverse rotation of the 4.2. Kinematic Analysis of a Stride Length Adjusting motor for stride length adjusting. Mechanism. The closed loop of the stride length adjusting mechanism is given as O O O O O (see Figure 3). The kine- 4.3. Kinematic Analysis of a Stride Length Mechanism. The 1 7 3 2 1 matic relations of the stride length adjusting mechanism can stride length mechanism is defined by O NN QRO O (see 1 6 1 be obtained by solving the following equation: Figure 3). Two closed loops, O NN QPO O and 1 7 1 O O PQO O , can be obtained by considering the stride 1 7 6 1 3 length adjusting mechanism. From the closed loop O NN X − X cos α + Y − Y sin α = l ± s , ð2Þ O O 1 O O 1 0 7 2 7 2 2π QPO O , we have 7 1 ! ! ! ! ! ! ð3Þ ′ ′ O N +NN +N Q =O O +O P + PQ , 1 1 7 7 j θ +φ +β +π/2 jðÞ θ +φ ðÞ jðÞ θ +φ +β +π jðÞ α +π/2 jα 1 1 1 10 1 1 1 1 1 10 ð4Þ R e + he + e e = X + Y j + r e + Se , 1 2 O O p 7 7 π π R cos θ + φ + h cos θ + φ +β + + e cos θ + φ +β + π = X + r cos α + + S cos α , ðÞ ðÞ 1 1 1 1 1 10 2 1 1 10 O p 1 1 2 2 ð5Þ π π R sinðÞ θ + φ + h sin θ + φ +β + + e sinðÞ θ + φ +β + π = Y + r sin α + + S sin α : 1 1 1 1 2 1 1 O p 1 1 1 10 10 2 2 Further, from the closed loop O O PQO O , we have where β = −β SIGNðφ Þ and θ = π/2 + θ + β + φ + γ , 1 7 6 1 10 1 1 2 1 10 1 2 and θ denotes the input angle of the driving motor for the ! ! ! ! ! ! ð6Þ stride length. When structural parameters a , b , and c of O O +O P + PQ + QR =O O +O R, 1 1 1 1 7 7 1 6 6 the stride length cam are determined, the values of φ , h, S, j α +π/2 jα θ jθ ðÞ and θ can be obtained by combining the relations given in 1 1 2 6 X + Y j + r e + Se + l e = X + Y j + r e , O O p 1 O O 6 7 7 6 6 Table 1 with equations (5) and (8) and using the Newton iter- ð7Þ ation method. X + r cos α + + S cos α + l cos θ = X + r cos θ , 4.4. Kinematic Analysis of a Four-Bar Linkage. From the O p 1 1 1 2 O 6 6 7 6 closed loop O ABO O (see Figure 3), we have 1 4 1 Y + r sin α + + S sin α + l sin θ = Y + r sin θ , O p 1 1 1 2 O 6 6 7 6 ! ! ! ! ð9Þ ð8Þ O A + AB =O O +O B, 1 1 4 4 Applied Bionics and Biomechanics 7 j θ +γ jθ j θ +γ ðÞ ðÞ where θ and θ can be obtained from equation (11) by using 1 1 A 4 4 A 4 r e + r e = X + Y j + r e , ð10Þ 1 AB O O 4 4 4 the Newton iteration method. r cosðÞ θ + γ + r cos θ = X + r cosðÞ θ + γ , 1 1 AB A O 4 4 1 4 4.5. Motion Analysis of a Lifting Mechanism. The closed loop r sinðÞ θ + γ + r sin θ = Y + r sinðÞ θ + γ , 1 1 AB A O 4 4 1 4 of the lifting mechanism is O O KK O O (see Figure 3), 1 4 5 1 ð11Þ and based on it, the following equations can be obtained: ! ! ! ! ! ð12Þ ′ ′ O O +O K +KK +K O =O O , 1 4 4 5 1 5 jðÞ θ +φ jðÞ θ +φ +β +π/2 jðÞ θ +φ +β +π 4 4 4 4 4 40 4 40 ′ ð13Þ X + Y j + R e + h e + e e = X + Y j, O O 4 5 O O 4 4 5 5 X + R cosðÞ θ + φ + h cos θ + φ + β + + e cosðÞ θ + φ + β + π = X , O 4 4 4 4 4 40 5 4 4 40 O 4 5 ð14Þ Y + R sinðÞ θ + φ + h sin θ + φ + β + + e sinðÞ θ + φ + β + π = Y , O 4 4 4 5 4 O 4 4 40 4 40 4 5 ′ 4.6. Motion Analysis of a Leg Mechanism. The closed loop of where β = −β SIGNðφ Þ, and φ and h can be obtained 40 4 4 4 the leg mechanism is O O SUTO O (see Figure 3). From from equation (14) by using the Newton iteration method. 1 5 6 1 this closed loop, we have ! ! ! ! ! ! ð15Þ O O + O S =O O +O T + TU + US , 1 5 5 1 6 6 jθ j θ +γ jθ j θ +θ −π ðÞ ðÞ 5 6 8 U 8 ð16Þ X + Y j + r e = X + Y j + r e + r e + le , O O 5 O O 7 8 5 5 6 6 X + r cos θ = X + r cos θ + γ + r cos θ + l cos θ + θ − π , ðÞ ðÞ O 5 5 O 7 6 6 8 8 U 8 5 6 ð17Þ Y + r sin θ = Y + r sin θ + γ + r sin θ + l sin θ + θ − π , ðÞ ðÞ O 5 5 O 7 6 6 8 8 U 8 5 6 where θ =3π/2 + θ + β + φ − γ , and θ and l can be 5. Analysis and Discussion of Pressure 5 4 8 40 4 5 obtained from equation (17) also by using the Newton itera- Angle Performance tion method. The pressure angle refers to a sharp angle formed between the normal on the cam profile on the contact point and the 4.7. Trajectory Coordinate of Leg End-Point. Following the velocity direction of the corresponding contact point of the schematic of the single-leg walking mechanism presented in follower. In the following, the pressure angle of the stride Figure 3, and solving equations (5), (8), (11), (14), and (17), length cam in the cam-linkage mechanism is discussed. the parameters l , α , θ , h, S, θ , and θ can be calculated, 7 1 6 5 8 The calculation model of the pressure angle of the stride and the trajectory coordinates of the leg end-point can be length cam is illustrated in Figure 6. According to the obtained by three-center theorem of the velocity instantaneous center, the three velocity instantaneous centers of three adjacent components must be on the same straight line. If the velocity X = X + r cosðÞ θ + γ + r cos θ + S cosðÞ π + θ − θ , W O 7 6 6 8 8 8 8 P instantaneous centers of two adjacent component pairs are Y = Y + r sinðÞ θ + γ + r sin θ + S sinðÞ π + θ − θ , determined, the velocity instantaneous center of another pair W O 7 6 6 8 8 8 8 P of components can be obtained based on the mentioned the- ð18Þ orem. It can be determined that point P is the velocity instantaneous center of the stride length fork that touches the stride length cam (Figure 6). where X and Y are the trajectory coordinates of the leg W W The pressure angle α can be calculated by setting up the end-point W in the horizontal and vertical directions, respectively. loop equations. From the closed loop O PQP O O , we have 7 6 7 8 Applied Bionics and Biomechanics ! ! ! ! ! ð19Þ ′ ′ O O +O P =O P + PQ +QP , 7 6 6 7 jθ jðÞ α +π/2 jðÞ α +π jðÞ α +3π/2 6 1 1 1 ð20Þ X − X + Y − Y j + l e = r e + Se + l e , O O O O 4 P 3 6 7 6 7 π 3π X − X + l cos θ = r cos α + + S cos α + π + l cos α + , ðÞ O O 4 6 P 1 1 3 1 < 6 7 2 2 ð21Þ π 3π : Y − Y + l sin θ = r sin α + + S sinðÞ α + π + l sin α + : O O 4 6 P 1 1 3 1 6 7 2 2 the rectangular swing rod, and the obtained results are pre- From the closed loop O O P NO , we have 1 6 1 sented in Figures 7–9. ! ! As can be seen in Figure 9, when α increased, the height ! ! ð22Þ ′ ′ O O +O P +P N =O N, 1 6 6 1 of the leg end-point trajectory changed significantly. ° ° ° The inclination angles of 12.5 , 14.5 , and 18.5 were used jθ jθ jðÞ θ +φ 6 N 1 ð23Þ X + Y j + l e + l e = R e , to analyze and explain the changes in the leg end-point tra- O O 4 2 1 6 6 jectory and the body trajectory, and analysis results are pre- X + l cos θ + l cos θ = R cos θ + φ , ðÞ sented in Figures 10–12. O 4 6 2 N 1 1 1 ð24Þ As can be seen in Figure 10, at the inclination angle of Y + l sin θ + l sin θ = R sin θ + φ : ðÞ ° O 4 6 2 N 1 1 1 12.5 , the maximum lifting height of the leg end-point W was 47.72 mm and the maximum horizontal moving distance By solving equations (21) and (24), l , l , l , and θ can be 2 3 4 N was 506.33 mm. For the robot body, the maximum change in obtained. Then, it holds that height was 67.76 mm, the maximum horizontal moving dis- tance was 631.22 mm, and the stride length of this kind of α = θ + −ðÞ θ + φ + β , ð25Þ N 1 1 10 gait was 417.18 mm. The results obtained at the inclination angle of 14.5 are presented in Figure 11, where it can be seen that the where β = −β SIGNðφ Þ. 10 1 1 maximum lifting height of the leg end-point W was 139.68 mm and the maximum horizontal moving distance 6. Calculation Example and Simulation was 611.79 mm. For the robot body, the maximum change in height was 67.13 mm, the maximum horizontal moving In this section, a calculation example and a motion simula- tion are given. The structural parameters of the designed distance was 555.32 mm, and the stride length of this kind single-leg system were as follows. Note that all units are of gait was 548.59 mm. Lastly, the results obtained at the inclination angle of 18.5 given in mm. The center O of the stride length cam was used as the origin of the coordinate system (see Figure 3), are shown in Figure 12, where it can be seen that the maxi- mum lifting height of W was 1293.14 mm and the maximum and the coordinates of the points were as follows: O (-360, -753), O (-460, -720), O (383, -531), O (-114, -654), and horizontal moving distance was 852.04 mm. For the robot 2 6 4 body, the maximum change in height was 30.62 mm, the max- O (-433, -923). The lengths where as follows: r = 185 5 p imum horizontal moving distance was 232.85 mm, and the (length of O P), l =240 (length of QR), e =165 (single-side 7 1 2 stride length of this kind of gait was 226.83 mm. Based on width of the stride length fork), r =44 (length of BO ), r = 4 4 1 the obtained results it can be concluded that when this kind 35 (length of O A), r =655 (length of AB), r =474 (length 1 AB 7 of leg end-point trajectory (shown in Figure 12) is used for of O T), r =95 (length of RO ), r = 100 (length of TU), 6 6 6 8 the gait of the bionic horse robot, a large lifting height of the S = 1081 (distance between point U and point W), r = 356 8 5 leg can be achieved, but the moving distance of the body will (length of O S), and e = 120 (single-side width of the stride 5 5 be small, which is similar to the horse strolling in situ. height fork). Additionally, the angles were γ = 180 , γ = 1 2 ° ° ° ° ° ° Generally speaking, at α =14:5 , this leg end-point tra- 40 , γ =62 , γ =40 , γ = −8 , θ = 120 , and θ = 125 . 1 5 6 4 U P jectory (shown in Figure 11) for the gait of the bionic horse The structural parameters of the stride cam were a = 220, robot would be a better choice. b = 160, and c =25, while those of the stride height cam 1 1 The pressure angles of the stride length cam mechanism were a = 220, b = 220, and c =10. MATLAB software was 2 2 2 during the forward gait were also determined, and the used for the development of the motion analysis system for obtained values are presented graphically in Figure 13. In the bionic horse. Figure 13, the rotation angle of the stride length cam is shown 6.1. Calculation Example. Based on the kinematic model of on the horizontal axis and the inclination angle is presented on the vertical axis; different line types are used to show the the bionic horse robot and the related input parameters, the change in the motion trajectory and the pressure angle for pressure angle for different values of the inclination angle. one cycle was calculated. Several trajectories of the leg end- In general, the maximum allowed pressure angle was 45 . point were obtained by changing the inclination angle α of Also, as can be seen in Figure 13, the maximum angle of the 1 r Applied Bionics and Biomechanics 9 Table 1: Motion parameters of the constant-breadth three-center cam. Absolute value of input anglejj φ Output angle β Radius vector R −1 jj φ ≤ sin 0 2a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a − b /4 a θ a ⋅ sinðÞ φSIGNφ − β −ðÞ θ /2 −1 x x b π −1 −1 sin ⋅ sin φSIGNφ − sin < φ ≤ + tan jj a + c 2 sin β 2a 2 a + c − b/2 ðÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a θ a − b /4 a ⋅ sinðÞ φSIGNφ − β +ðÞ θ /2 −1 x x π b −1 −1 sin ⋅ sin φSIGNφ + + tan < φ ≤ π‐sin jj a + c‐b 2 sin β 2 a + c − b/2 2a ðÞ −1 2a + c − b π − sin <jj φ ≤ π 0 2a −700 R −800 ' −900 −1000 −1100 1 1 훼 −1200 −1300 −1400 −1500 N' −1600 −1700 −1800 −1900 S −2000 −2100 −2200 휃 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 휃 (°) P' 훼 = 12.5° 훼 = 16.5° 1 1 훼 = 13.5° 훼 = 17.5° 훼 = 14.5° 훼 = 18.5° 1 1 훼 = 15.5° Figure 6: Calculation model of the pressure angle of the stride Figure 8: The vertical direction trajectories of the leg end-point for length cam. moving forward at different inclination angles. −800 −900 −1000 −1100 −1200 −1300 −1400 −1500 −1600 −1700 −1800 −1900 −2000 −2100 −2200 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 200 300 400 500 600 700 800 900 1000 휃 (°) X (mm) 훼 = 12.5° 훼 = 16.5° 훼 = 12.5° 훼 = 16.5° 1 1 1 1 훼 = 13.5° 훼 = 17.5° 훼 = 13.5° 훼 = 17.5° 1 1 1 1 훼 = 14.5° 훼 = 18.5° 훼 = 14.5° 훼 = 18.5° 1 1 1 1 훼 = 15.5° 훼 = 15.5° Ground line 1 1 Figure 7: The horizontal direction trajectories of the leg end-point Figure 9: The trajectories of the leg end-point for moving forward for moving forward at different inclination angles. at different inclination angles. X (mm) Y (mm) Y (mm) w w 10 Applied Bionics and Biomechanics −720 −700 40 W −740 −800 W 60 −760 200 300 400 500 600 700 800 900 1000 1100 X (mm) −780 w W −800 −800 −900 200 300 400 500 600 700 800 900 1000 1100 −1000 −1100 X (mm) −1200 −2020 −1300 −2040 −1400 2 −1500 −2060 −1600 −2080 −1700 1 W 4 2 −1800 Ground line −2100 Ground line 3 W −1900 −2120 W −2000 1 W 6 4 −2100 W W −2140 5 −2200 200 300 400 500 600 700 800 900 1000 1100 200 300 400 500 600 700 800 900 1000 1100 X (mm) X (mm) Figure 12: The trajectories of the leg end-point (bottom) and the Figure 10: The trajectories of the leg end-point (bottom) and the body (top) at α =18:5 . body (top) at α =12:5 . −740 −760 −780 10 22 −800 200 300 400 500 600 700 800 900 1000 1100 X (mm) −1920 −1940 W 17 −1960 −1980 −2000 −2020 −2040 −2060 −2080 −2100 10 W 5 6 Ground line −2120 9 −2140 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 휃 (°) 200 300 400 500 600 700 800 900 1000 1100 X (mm) 훼 = 12.5° 훼 = 16.5° 1 1 훼 = 13.5° 훼 = 17.5° 1 1 Figure 11: The trajectories of the leg end-point (bottom) and the 훼 = 14.5° 훼 = 18.5° 1 1 body (top) at α =14:5 . 훼 = 15.5° Figure 13: Pressure angles of the stride length cam mechanism stride length cam mechanism was less than 26 , which was during the forward gait for different inclination angles. within the allowed range. The forward motion simulation for one leg is shown in Figure 15. The motion trajectory of the leg and the pressure 6.2. Motion Simulation. In order to validate the calculated values of the motion trajectory and pressure angle of the angle of the cam were automatically exported by the simula- bionic horse, a structural model of the robot was developed, tion software. In Figure 16, the trajectory of the leg end-point W during and its motion was simulated in Solidworks software. The ° ° model of the forward leg motion at α =14:5 of the bionic the forward gait at α =14:5 obtained by both the motion simulation (solid line) and the theoretical calculation (dotted horse robot in the simulation software is presented in Figures 14 and 15. line) is shown. As can be seen in Figure 16, the maximum dif- A forward motion cycle simulation for the robot entirety ferences between the motion simulation trajectory and theo- is shown in Figure 14, (a) shows the initial position of the retically calculated trajectory were 4.60 mm in the x-direction robot, (b) shows the forward movement of the first leg, (c) and 2.15 mm in the y-direction. In Figure 17, the pressure angle curve of the stride length shows the forward movement of the second leg, (d) shows the forward movement of the third leg, (e) shows the forward cam during the forward gait at α =14:5 obtained by both movement of the fourth leg, and (f) shows the movement of the motion simulation (solid line) and the theoretical calcula- the robot body completing a forward gait. tion (dotted line) is shown. The maximum differences in the Y (mm) Y (mm) Y (mm) Y (mm) w w w w (°) Y (mm) Y (mm) w w Applied Bionics and Biomechanics 11 (a) (b) (c) (d) (e) (f) Figure 14: A forward motion cycle simulation for the robot entirety at α =14:5 in Solidworks software. Figure 15: The forward leg motion simulation at α =14:5 in Solidworks software. −1920 −1940 −1960 −1980 W 15 −2000 −2020 −2040 W 13 −2060 W 4 −2080 −2100 W Ground line −2120 −2140 200 300 400 500 600 700 800 900 1000 1100 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 X (mm) 휃 (°) Theoretical value Theoretical value Actual value Actual value Figure 16: The trajectory of the leg end-point during the forward Figure 17: The pressure angle of the stride length cam during the ° ° gait at α =14:5 . forward gait at α =14:5 . 1 1 Y (mm) (°) 12 Applied Bionics and Biomechanics pressure angle between the motion simulation and the theo- IEEE International Conference on Robotics and Automation, pp. 4736–4741, Anchorage, AK, USA, 2010. retical calculation were 0.21 . Based on the results demonstrated in Figures 16 and [3] W. Guobiao, C. Diansheng, C. Kewei, and Z. 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