CN102579137A - Parallel surgical manipulator capable of horizontally moving three-dimensionally and rotating one-dimensionally - Google Patents

Parallel surgical manipulator capable of horizontally moving three-dimensionally and rotating one-dimensionally Download PDF

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CN102579137A
CN102579137A CN2012100848721A CN201210084872A CN102579137A CN 102579137 A CN102579137 A CN 102579137A CN 2012100848721 A CN2012100848721 A CN 2012100848721A CN 201210084872 A CN201210084872 A CN 201210084872A CN 102579137 A CN102579137 A CN 102579137A
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connecting rod
pair
chain
turntable
parallel
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CN102579137B (en
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蔡敢为
潘宇晨
王红州
高德中
张�林
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Guangxi University
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Guangxi University
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Abstract

The invention relates to a parallel surgical manipulator capable of horizontally moving three-dimensionally and rotating one-dimensionally, which comprises a first RPRPR closed-loop sub-chain, a second RPRPR closed-loop sub-chain and an actuating mechanism sub-chain. The first RPRPR closed-loop sub-chain and the second RPRPR closed-loop sub-chain are located in the same plane. The first RPRPR closed-loop sub-chain can control a first rotary table to move with two degrees of freedom in the plane, and the second RPRPR closed-loop sub-chain can control a second rotary table to move with two degrees of freedom in the plane. A movable platform can move with four degrees of freedom in a space due to movement of the first rotary table and movement of the second rotary table. The two closed-loop sub-chains are used for controlling movement of the two rotary tables in the plane, and accordingly the movable platform can move with four degrees of freedom in the space. The parallel surgical manipulator has the advantages of compact structure and simplicity in control. All driving rods are connected onto a framework and can be light rods, inertia of mechanism kinematics is low, and dynamics performances are good.

Description

一种可实现三维平动一维转动的并联手术机械手A parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation

技术领域 technical field

本发明涉及机器人领域,特别是一种可实现三维平动一维转动的并联手术机械手。The invention relates to the field of robots, in particular to a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation.

背景技术 Background technique

并联机构是一种动平台和定平台通过至少两个独立的运动链相连接,有两个或两个以上自由度,且以并联方式驱动的机构,具有精度较高、结构紧凑、刚度高、承载能力大强等优点,在运动模拟器、并联机床、微操作机器人、抓取机器人等方面取得了较好的应用。但现有的并联机构基本都是分支不含闭环的,对于自由度数大于支链数的并联机构,有些支链必须设置两个或两个以上的原动件,这就不得不将电机安装在支链的关节处,从而导致机构刚性差、惯量大、关节误差累积等问题,机构动力学性能较差,控制困难。手术用机器人是要求非常高的高精准仪器,传统的串联机器人难以达到其使用要求,采用并联机构可满足手术用机器人的精确运动的要求。The parallel mechanism is a mechanism that connects the moving platform and the fixed platform through at least two independent kinematic chains, has two or more degrees of freedom, and is driven in parallel. It has high precision, compact structure, high rigidity, With the advantages of strong carrying capacity and other advantages, it has achieved good applications in motion simulators, parallel machine tools, micro-manipulation robots, and grasping robots. However, the existing parallel mechanisms basically have branches without closed loops. For parallel mechanisms with degrees of freedom greater than the number of branch chains, some branch chains must be equipped with two or more driving parts, which requires the motor to be installed in the Joints of branch chains, resulting in problems such as poor rigidity of the mechanism, large inertia, accumulation of joint errors, poor dynamic performance of the mechanism, and difficult control. Surgical robots are high-precision instruments with very high requirements. It is difficult for traditional serial robots to meet their requirements. The use of parallel mechanisms can meet the requirements of precise movement of surgical robots.

发明内容 Contents of the invention

本发明的目的在于提供一种可实现三维平动一维转动的并联手术机械手,解决传统自由度数大于支链数的并联手术机械手刚性差、惯量大、关节误差累积等问题。The purpose of the present invention is to provide a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation, which solves the problems of poor rigidity, large inertia, and joint error accumulation of traditional parallel surgical manipulators whose number of degrees of freedom is greater than the number of branches.

本发明通过以下技术方案达到上述目的:一种可实现三维平动一维转动的并联手术机械手,包括第一RPRPR闭环子链、第二RPRPR闭环子链和执行机构子链,其机构和链接方式为:The present invention achieves the above object through the following technical solutions: a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation, including the first RPRPR closed-loop sub-chain, the second RRPPR closed-loop sub-chain and the actuator sub-chain, its mechanism and linking mode for:

所述第一RPRPR闭环子链由第一连杆、第二连杆、第三连杆、第四连杆以及机架连接而成,第一连杆通过第一转动副连接到机架上,第一连杆另一端通过第一移动副与第二连杆连接,第二连杆另一端通过第二转动副与第三连杆连接,第三连杆另一端通过第二移动副与第四连杆连接,第四连杆另一端通过第三转动副连接到机架上,第一连杆由第一伺服电机驱动,第四连杆由第二伺服电机驱动,The first RPRPR closed-loop sub-chain is formed by connecting the first connecting rod, the second connecting rod, the third connecting rod, the fourth connecting rod and the frame, the first connecting rod is connected to the frame through the first rotating pair, The other end of the first connecting rod is connected to the second connecting rod through the first moving pair, the other end of the second connecting rod is connected to the third connecting rod through the second rotating pair, and the other end of the third connecting rod is connected to the fourth connecting rod through the second moving pair. The other end of the fourth connecting rod is connected to the frame through the third rotating pair, the first connecting rod is driven by the first servo motor, and the fourth connecting rod is driven by the second servo motor.

所述第二RPRPR闭环子链由第五连杆、第六连杆、第七连杆、第八连杆以及机架连接而成,第五连杆通过第四转动副连接到机架上,第五连杆另一端通过第三移动副与第六连杆连接,第六连杆另一端通过第五转动副与第七连杆连接,第七连杆另一端通过第四移动副与第八连杆连接,第八连杆另一端通过第六转动副连接到机架上,第五连杆由第三伺服电机驱动,第八连杆由第四伺服电机驱动,The second RPRPR closed-loop sub-chain is formed by connecting the fifth connecting rod, the sixth connecting rod, the seventh connecting rod, the eighth connecting rod and the frame, and the fifth connecting rod is connected to the frame through the fourth rotating pair, The other end of the fifth connecting rod is connected to the sixth connecting rod through the third moving pair, the other end of the sixth connecting rod is connected to the seventh connecting rod through the fifth rotating pair, and the other end of the seventh connecting rod is connected to the eighth connecting rod through the fourth moving pair. The connecting rod is connected, the other end of the eighth connecting rod is connected to the frame through the sixth rotating pair, the fifth connecting rod is driven by the third servo motor, the eighth connecting rod is driven by the fourth servo motor,

所述执行机构子链由第一转台、第二转台、第九连杆、第十连杆、第十一连杆、第十二连杆以及动平台连接而成,第一转台通过第二转动副与第二连杆连接,第一转台通过第七转动副与第九连杆连接,第一转台通过第八转动副与第十连杆连接,第九连杆另一端通过第九转动副与动平台连接,第十连杆另一端通过第十转动副与动平台连接,第二转台通过五转动副与第六连杆连接,第二转台通过第十一转动副与第十一连杆连接,第二转台通过第十二转动副与第十二连杆连接,第十一连杆另一端通过第九转动副与动平台和第九连杆连接,第十二连杆另一端通过第十三转动副与动平台连接。第一转台和第二转台在平面内的运动可实现动平台在空间的四自由度运动,The sub-chain of the actuator is connected by the first turntable, the second turntable, the ninth connecting rod, the tenth connecting rod, the eleventh connecting rod, the twelfth connecting rod and the moving platform. The pair is connected with the second connecting rod, the first turntable is connected with the ninth connecting rod through the seventh rotating pair, the first turntable is connected with the tenth connecting rod through the eighth rotating pair, and the other end of the ninth connecting rod is connected with the ninth connecting rod through the ninth rotating pair. The other end of the tenth connecting rod is connected to the moving platform through the tenth revolving pair, the second turntable is connected to the sixth connecting rod through the fifth revolving pair, and the second turntable is connected to the eleventh connecting rod through the eleventh revolving pair , the second turntable is connected with the twelfth connecting rod through the twelfth rotating pair, the other end of the eleventh connecting rod is connected with the moving platform and the ninth connecting rod through the ninth rotating pair, and the other end of the twelfth connecting rod is connected through the tenth connecting rod The three revolving pairs are connected with the moving platform. The movement of the first turntable and the second turntable in the plane can realize the four-degree-of-freedom movement of the moving platform in space,

所述第一转动副、第二转动副、第三转动副、第四转动副、第五转动副、第六转动副的旋转轴线相互平行,第七转动副、第八转动副、第九转动副、第十转动副、第十一转动副、第十二转动副、第十三转动副的旋转轴线相互平行,第九连杆与第十连杆等长平行,第十一连杆与第十二连杆等长平行。The rotation axes of the first, second, third, fourth, fifth, and sixth rotation pairs are parallel to each other, and the seventh, eighth, and ninth rotation pairs are parallel to each other. The axes of rotation of the pair, the tenth revolving pair, the eleventh revolving pair, the twelfth revolving pair, and the thirteenth revolving pair are parallel to each other, the ninth connecting rod is parallel to the tenth connecting rod, and the eleventh connecting rod is parallel to the tenth connecting rod. Twelve connecting rods are equal in length and parallel.

本发明的突出优点在于:The outstanding advantages of the present invention are:

1、驱动电机均安装在机架上,杆件都可以做成轻杆,有效降低机构重量,机构刚性好、惯量小,动力学性能好;1. The driving motors are all installed on the frame, and the rods can be made into light rods, which can effectively reduce the weight of the mechanism, with good rigidity, small inertia and good dynamic performance;

2、通过两个对称结构的闭环子链控制动平台运动输出,机构运动学正、逆问题求解容易,控制方便;2. The motion output of the braking platform is controlled by two closed-loop sub-chains of symmetrical structure, so that the forward and inverse problems of mechanism kinematics are easy to solve and the control is convenient;

3、通过在动平台上安装手术刀等手术工具来完成手术动作任务。3. Complete surgical action tasks by installing surgical tools such as scalpels on the moving platform.

附图说明 Description of drawings

图1为本发明所述一种可实现三维平动一维转动的并联手术机械手的第一结构示意图。FIG. 1 is a schematic diagram of the first structure of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

图2为本发明所述一种可实现三维平动一维转动的并联手术机械手的第二结构示意图。Fig. 2 is a schematic diagram of the second structure of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

图3为本发明所述一种可实现三维平动一维转动的并联手术机械手的第一工作示意图。Fig. 3 is a first working schematic diagram of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

图4为本发明所述一种可实现三维平动一维转动的并联手术机械手的第二工作示意图。Fig. 4 is a second working schematic diagram of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

图5为本发明所述一种可实现三维平动一维转动的并联手术机械手的第三工作示意图。Fig. 5 is a third working schematic diagram of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

图6为本发明所述一种可实现三维平动一维转动的并联手术机械手的第四工作示意图。Fig. 6 is a fourth working schematic diagram of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

图7为本发明所述一种可实现三维平动一维转动的并联手术机械手的第五工作示意图。Fig. 7 is a fifth working diagram of a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation according to the present invention.

具体实施方式 Detailed ways

下面结合附图及实施例对本发明的技术方案作进一步说明。The technical solutions of the present invention will be further described below in conjunction with the accompanying drawings and embodiments.

对照图1和图2,一种可实现三维平动一维转动的并联手术机械手,包括第一RPRPR闭环子链、第二RPRPR闭环子链和执行机构子链。Referring to Fig. 1 and Fig. 2, a parallel surgical manipulator capable of three-dimensional translation and one-dimensional rotation includes a first RRPPR closed-loop sub-chain, a second RRPPR closed-loop sub-chain and an actuator sub-chain.

所述第一RPRPR闭环子链由第一连杆3、第二连杆4、第三连杆8、第四连杆9以及机架1连接而成,第一连杆3通过第一转动副2连接到机架1上,第一连杆3另一端通过第一移动副30与第二连杆4连接,第二连杆4另一端通过第二转动副28与第三连杆8连接,第三连杆8另一端通过第二移动副与第四连杆9连接,第四连杆9另一端通过第三转动副10连接到机架1上,第一连杆3由第一伺服电机34驱动,第四连杆9由第二伺服电机35驱动。The first RRPPR closed-loop sub-chain is formed by connecting the first connecting rod 3, the second connecting rod 4, the third connecting rod 8, the fourth connecting rod 9 and the frame 1, and the first connecting rod 3 passes through the first rotating pair 2 is connected to the frame 1, the other end of the first connecting rod 3 is connected to the second connecting rod 4 through the first moving pair 30, and the other end of the second connecting rod 4 is connected to the third connecting rod 8 through the second rotating pair 28, The other end of the third connecting rod 8 is connected to the fourth connecting rod 9 through the second moving pair, the other end of the fourth connecting rod 9 is connected to the frame 1 through the third rotating pair 10, and the first connecting rod 3 is driven by the first servo motor 34 drives, and the fourth connecting rod 9 is driven by the second servo motor 35.

所述第二RPRPR闭环子链由第五连杆20、第六连杆21、第七连杆25、第八连杆26以及机架1连接而成,第五连杆20通过第四转动副19连接到机架1上,第五连杆另一端通过第三移动副32与第六连杆21连接,第六连杆21另一端通过第五转动副29与第七连杆25连接,第七连杆25另一端通过第四移动副33与第八连杆26连接,第八连杆26另一端通过第六转动副27连接到机架1上,第五连杆20由第三伺服电机36驱动,第八连杆26由第四伺服电机37驱动。The second RRPPR closed-loop sub-chain is formed by connecting the fifth connecting rod 20, the sixth connecting rod 21, the seventh connecting rod 25, the eighth connecting rod 26 and the frame 1, and the fifth connecting rod 20 passes through the fourth rotating pair 19 is connected to the frame 1, the other end of the fifth connecting rod is connected with the sixth connecting rod 21 through the third moving pair 32, and the other end of the sixth connecting rod 21 is connected with the seventh connecting rod 25 through the fifth rotating pair 29, the second The other end of the seven connecting rod 25 is connected to the eighth connecting rod 26 through the fourth moving pair 33, the other end of the eighth connecting rod 26 is connected to the frame 1 through the sixth rotating pair 27, and the fifth connecting rod 20 is driven by the third servo motor 36 drives, and the eighth connecting rod 26 is driven by the fourth servo motor 37.

所述执行机构子链由第一转台5、第二转台24、第九连杆12、第十连杆11、第十一连杆18、第十二连杆17以及动平台14连接而成,第一转台5通过第二转动副28与第二连杆4连接,第一转台5通过第七转动副7与第九连杆12连接,第一转台5通过第八转动副6与第十连杆11连接,第九连杆12另一端通过第九转动副15与动平台14连接,第十连杆11另一端通过第十转动副13与动平台14连接,第二转台24通过五转动副29与第六连杆21连接,第二转台24通过第十一转动副22与第十一连杆18连接,第二转台24通过第十二转动副23与第十二连杆17连接,第十一连杆18另一端通过第九转动副15与动平台14和第九连杆12连接,第十二连杆17另一端通过第十三转动副16与动平台15连接。第一转台5和第二转台24在平面内的运动可实现动平台14在空间的4自由度运动。The sub-chain of the actuator is formed by connecting the first turntable 5, the second turntable 24, the ninth connecting rod 12, the tenth connecting rod 11, the eleventh connecting rod 18, the twelfth connecting rod 17 and the moving platform 14, The first turntable 5 is connected with the second connecting rod 4 through the second turning pair 28, the first turntable 5 is connected with the ninth connecting rod 12 through the seventh turning pair 7, and the first turntable 5 is connected with the tenth connecting rod through the eighth turning pair 6. The rod 11 is connected, the other end of the ninth connecting rod 12 is connected with the moving platform 14 through the ninth rotating pair 15, the other end of the tenth connecting rod 11 is connected with the moving platform 14 through the tenth rotating pair 13, and the second turntable 24 is connected through the fifth rotating pair 29 is connected with the sixth connecting rod 21, the second turntable 24 is connected with the eleventh connecting rod 18 through the eleventh turning pair 22, the second turntable 24 is connected with the twelfth connecting rod 17 through the twelfth turning pair 23, and the second turntable 24 is connected with the twelfth connecting rod 17 through the twelfth turning pair 23, The other end of the eleventh connecting rod 18 is connected to the moving platform 14 and the ninth connecting rod 12 through the ninth rotating pair 15 , and the other end of the twelfth connecting rod 17 is connected to the moving platform 15 through the thirteenth rotating pair 16 . The movement of the first turntable 5 and the second turntable 24 in the plane can realize the 4-degree-of-freedom movement of the moving platform 14 in space.

所述第一转动副2、第二转动副28、第三转动副10、第四转动副19、第五转动副29、第六转动副27的旋转轴线相互平行,第七转动副7、第八转动副6、第九转动副15、第十转动副13、第十一转动副22、第十二转动副23、第十三转动副16的旋转轴线相互平行,第九连杆12与第十连杆11等长平行,第十一连杆18与第十二连杆17等长平行。The rotation axes of the first rotating pair 2, the second rotating pair 28, the third rotating pair 10, the fourth rotating pair 19, the fifth rotating pair 29, and the sixth rotating pair 27 are parallel to each other, and the seventh rotating pair 7, the sixth rotating pair The axes of rotation of the eighth revolving pair 6, the ninth revolving pair 15, the tenth revolving pair 13, the eleventh revolving pair 22, the twelfth revolving pair 23, and the thirteenth revolving pair 16 are parallel to each other, and the ninth connecting rod 12 and the The ten connecting rods 11 are equal in length and parallel, and the eleventh connecting rod 18 is equal in length and parallel to the twelfth connecting rod 17 .

对照图3和图4,第一转台4和第二转台21在运动空间内运动可实现动平台13的上下平移。Referring to FIG. 3 and FIG. 4 , the movement of the first turntable 4 and the second turntable 21 in the movement space can realize the vertical translation of the movable platform 13 .

对照图5,第一转台4和第二转台21在运动空间内运动可实现动平台13绕自身轴线的一维整周转动。Referring to FIG. 5 , the movement of the first turntable 4 and the second turntable 21 in the movement space can realize the one-dimensional complete rotation of the moving platform 13 around its own axis.

对照图6,第一转台4和第二转台21在运动空间内运动可实现动平台13的前后平移。Referring to FIG. 6 , the movement of the first turntable 4 and the second turntable 21 in the movement space can realize the forward and backward translation of the movable platform 13 .

对照图7,第一转台4和第二转台21在运动空间内运动可实现动平台13的左右平移。Referring to FIG. 7 , the movement of the first turntable 4 and the second turntable 21 in the movement space can realize the left-right translation of the movable platform 13 .

Claims (1)

1.一种可实现三维平动一维转动的并联手术机械手,包括第一RPRPR闭环子链、第二RPRPR闭环子链和执行机构子链,其特征在于,其机构和链接方式为:1. A parallel surgical manipulator capable of realizing three-dimensional translation and one-dimensional rotation, comprising the first RPRPR closed-loop sub-chain, the second RPRPR closed-loop sub-chain and the actuator sub-chain, characterized in that its mechanism and linking methods are: 所述第一RPRPR闭环子链由第一连杆、第二连杆、第三连杆、第四连杆以及机架连接而成,第一连杆通过第一转动副连接到机架上,第一连杆另一端通过第一移动副与第二连杆连接,第二连杆另一端通过第二转动副与第三连杆连接,第三连杆另一端通过第二移动副与第四连杆连接,第四连杆另一端通过第三转动副连接到机架上,第一连杆由第一伺服电机驱动,第四连杆由第二伺服电机驱动,The first RPRPR closed-loop sub-chain is formed by connecting the first connecting rod, the second connecting rod, the third connecting rod, the fourth connecting rod and the frame, the first connecting rod is connected to the frame through the first rotating pair, The other end of the first connecting rod is connected to the second connecting rod through the first moving pair, the other end of the second connecting rod is connected to the third connecting rod through the second rotating pair, and the other end of the third connecting rod is connected to the fourth connecting rod through the second moving pair. The other end of the fourth connecting rod is connected to the frame through the third rotating pair, the first connecting rod is driven by the first servo motor, and the fourth connecting rod is driven by the second servo motor. 所述第二RPRPR闭环子链由第五连杆、第六连杆、第七连杆、第八连杆以及机架连接而成,第五连杆通过第四转动副连接到机架上,第五连杆另一端通过第三移动副与第六连杆连接,第六连杆另一端通过第五转动副与第七连杆连接,第七连杆另一端通过第四移动副与第八连杆连接,第八连杆另一端通过第六转动副连接到机架上,第五连杆由第三伺服电机驱动,第八连杆由第四伺服电机驱动,The second RPRPR closed-loop sub-chain is formed by connecting the fifth connecting rod, the sixth connecting rod, the seventh connecting rod, the eighth connecting rod and the frame, and the fifth connecting rod is connected to the frame through the fourth rotating pair, The other end of the fifth connecting rod is connected to the sixth connecting rod through the third moving pair, the other end of the sixth connecting rod is connected to the seventh connecting rod through the fifth rotating pair, and the other end of the seventh connecting rod is connected to the eighth connecting rod through the fourth moving pair. The connecting rod is connected, the other end of the eighth connecting rod is connected to the frame through the sixth rotating pair, the fifth connecting rod is driven by the third servo motor, the eighth connecting rod is driven by the fourth servo motor, 所述执行机构子链由第一转台、第二转台、第九连杆、第十连杆、第十一连杆、第十二连杆以及动平台连接而成,第一转台通过第二转动副与第二连杆连接,第一转台通过第七转动副与第九连杆连接,第一转台通过第八转动副与第十连杆连接,第九连杆另一端通过第九转动副与动平台连接,第十连杆另一端通过第十转动副与动平台连接,第二转台通过五转动副与第六连杆连接,第二转台通过第十一转动副与第十一连杆连接,第二转台通过第十二转动副与第十二连杆连接,第十一连杆另一端通过第九转动副与动平台和第九连杆连接,第十二连杆另一端通过第十三转动副与动平台连接。第一转台和第二转台在平面内的运动可实现动平台在空间的四自由度运动,The sub-chain of the actuator is connected by the first turntable, the second turntable, the ninth connecting rod, the tenth connecting rod, the eleventh connecting rod, the twelfth connecting rod and the moving platform. The pair is connected with the second connecting rod, the first turntable is connected with the ninth connecting rod through the seventh rotating pair, the first turntable is connected with the tenth connecting rod through the eighth rotating pair, and the other end of the ninth connecting rod is connected with the ninth connecting rod through the ninth rotating pair. The other end of the tenth connecting rod is connected to the moving platform through the tenth revolving pair, the second turntable is connected to the sixth connecting rod through the fifth revolving pair, and the second turntable is connected to the eleventh connecting rod through the eleventh revolving pair , the second turntable is connected with the twelfth connecting rod through the twelfth rotating pair, the other end of the eleventh connecting rod is connected with the moving platform and the ninth connecting rod through the ninth rotating pair, and the other end of the twelfth connecting rod is connected through the tenth connecting rod The three revolving pairs are connected with the moving platform. The movement of the first turntable and the second turntable in the plane can realize the four-degree-of-freedom movement of the moving platform in space, 所述第一转动副、第二转动副、第三转动副、第四转动副、第五转动副、第六转动副的旋转轴线相互平行,第七转动副、第八转动副、第九转动副、第十转动副、第十一转动副、第十二转动副、第十三转动副的旋转轴线相互平行,第九连杆与第十连杆等长平行,第十一连杆与第十二连杆等长平行。The rotation axes of the first, second, third, fourth, fifth, and sixth rotation pairs are parallel to each other, and the seventh, eighth, and ninth rotation pairs are parallel to each other. The axes of rotation of the pair, the tenth revolving pair, the eleventh revolving pair, the twelfth revolving pair, and the thirteenth revolving pair are parallel to each other, the ninth connecting rod is parallel to the tenth connecting rod, and the eleventh connecting rod is parallel to the tenth connecting rod. Twelve connecting rods are equal in length and parallel.
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