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1、Keywords关键词parallel manipulator; model-based control; dynamics; kinematics并联机器人；基于模型的控制；动态IC；运动学INTRODUCTION介绍Parallel manipulator has been extensively studied due to its high force-to-weight ratio and widespread application 1-2.并联机器人已被广泛研究，由于其高动力重量比和广泛的应用 1 - 2。6-DOF parallel manipulator is named S

2、tewart platform after Stewart illustrated the use of such parallel structure 3,it is also referred to as Gough-platform who presented the practical use of such a system 4.六自由度并联机器人取名斯图尔特平台后，斯图尔特说明使用这种平行结构 3，它也被称为高夫平台，高夫提出实际使用这种系统 4。Hunt 5 researched the kinematics of parallel manipulators based on s

3、crew theory and enumerated promising kinematics structures.亨特 5研究了并联机器人运动学基于螺旋理论和列举的有前途的运动学结构。Do and Yang 6 used the Newton-Euler approach to solve the inverse dynamics for Stewart platform assuming the joints as frictionless and legs asymmetrical.和阳 6 采用牛顿-欧拉法求解逆动力学假设斯图尔特平台关节摩擦和腿不对称。Nguyen et al. 7

4、 have developed a joint-space adaptive control scheme applied to an electromechanically driven Stewart platform using Lyapunov direct method.阮等人 7已经制定了一个空间自适应控制方案应用于机电驱动斯图尔特平台利用李雅普诺夫直接方法。Liu et al 8 discussed task-space control scheme for Stewart platform based milling cell.刘等人 8讨论工作空间控制方案的斯图尔特平台铣削单

5、元。Hatip and Ozgoren 9 developed a dynamic control strategy for Stewart platform. hatip和ozgoren 9 提出了动态控制策略斯图尔特平台Noriega et al. 10presented a neural network control scheme and showed its superiority over a kinematics control.诺列加等人 10提出了一个神经网络控制方案，在运动控制方面显示出其优越性Davliakos et al 11 developed operational

6、 error joint feedback control scheme embedding the forward kinematics in the feedback control loop for 6-DOF electrohydraulic Parallel manipulator platforms.davliakos等人 11 开发操作误差反馈控制方案嵌入关节运动学在反馈控制回路的六自由度并联机器人平台电液。Kim et al. 12 researched and applied a high speed tracking control for 6-DOF electric d

7、riven Stewart platform using an enhanced sliding mode control approach.基姆等人。 12研究和应用的一个高速度跟踪控制的六自由度电动驱动的斯图尔特平台的使用增强的滑模控制方法。A model-based controller for 6-DOF hydraulic driven parallel manipulator with symmetric joint locations is developed to reduce the effect of load variety of platform and elimina

8、te the steady state error of the control systems基于模型的控制器的六自由度液压驱动并联机器人的对称关节的位置来降低负载的影响多种平台和消除稳态误差的控制系统Multi-rigid body dynamics models are built considering the Gough-Stewart platform as 13 rigid body based on Kane method.多刚体动力学模型的建立，考虑到斯图尔特的平台有13刚体基于凯恩方法。The forward kinematics and inverse kinematic

9、s models are described with Newton-Raphson method and closed-form solution.正运动学和逆运动学模型描述的牛顿迭代法和解析解。The developed controller employs rigid body dynamic and yields the input current vector of the servovalve, the dynamic gravity term including the gravity of platform, load and hydraulic cylinders is us

10、ed to compensate the influence of gravity of parallel manipulator platform.该控制器采用刚体动力学和产量的输入电流矢量的伺服阀，动态重力项包括重力平台，负载和液压缸，用于补偿重力的影响，对并联机器人平台。In analytical, the steady state errors converge asymptotically to zero, independent of load variation.在分析，稳态误差渐近收敛于零，独立的负载变化。The model-based controller, PD contr

11、ol with gravity compensation, is developed to reduce the effect of load variety of platform and eliminate steady state error of hydraulic driven parallel manipulator.基于模型的控制器，控制重力补偿，以减少开发影响负载多种平台和消除稳态误差的液压驱动并联机器人。MATHEMATICAL MODEL数学模型The 6-DOF hydraulic driven parallel manipulator consist of a fixe

12、d base （down platform） and a moveable platform（upper platform） with six cylinders supporting it, all the cylinders are connected with movement platform and base with Hooke joints, as shown in Fig. 1.六自由度液压驱动并联机器人包括一个固定基地（下）和一个可移动的平台（平台）六缸支持它，所有气缸的运动平台和基地连接万向接头，如图1所示。Figure 1. Hydraulic driven 6-DOF

13、parallel manipulator 图1。液压驱动六自由度并联机器人A. Kinematics Model Kinematics is the science of motion that treats the subject without regard to the forces that cause it 13, the kinematics of 6-DOF parallel manipulator include inverse kinematics and forward kinematics, forward kinematics is used to solve the

14、generalized coordinates of upper platform with the length of leg as its input variable, the forward kinematics equation can be showed as j +1 = j + J l -1 （L 0 - L j ） （1）A.运动学模型运动学是运动科学，对待这个问题没有考虑到的力量，因为它 13，六自由度并联机器人的运动学逆运动学，包括运动学，运动学是用来解决广义坐标上平台与腿部的长度作为输入变量，正运动学方程可以显示如 j +1 = j + J l -1 （L 0 - L

15、j ）（1）where = （q1 , q 2 , q 3 , q 4 , q 5 , q 6 ） T is the 61 vector of the platform generalized coordinates, q1 , q 2 , q 3 , are the platform center of mass Cartesian coordinates, q 4 , q 5 , q 6 , are the platform Euler angles under Z-Y-X order, j is the iterative numbers, L0 is a measured length

16、 61 vector of leg of the platform, Lj is the 61 solving vector during the iterative calculation.在 = （q1 , q 2 , q 3 , q 4 , q 5 , q 6 ） T 是61向量的平台广义坐标，第1，2，3，是平台质心笛卡尔坐标，4，5，6，是欧拉的平台角下z-y-x秩序，是迭代次数，L0是一种测量长度61向量的腿的平台，Lj是61在迭代计算求解矢量。Inverse kinematics of parallel manipulator is different from serial m

17、anipulator, the length of leg of platform can be solved by closed-form solution, and it can be described byL = （ R *A - B） + t （2）where L is a 36 length matrix of leg of platform, R is a 33 rotation matrix from body coordinates to global coordinates, A is a 36 matrix of upper joints points, B is a 3

18、6 matrix of down joints points, and t is position 31 vector of platform.并联机器人逆运动学的不同长度的串联机器人，腿平台可以解决，封闭形式的解决方案，它可以描述L = （ R *A - B） + t （2）L是一个36长度矩阵腿平台，R是一个33旋转矩阵坐标的全球坐标，A是一个36矩阵的关节点上，B是一个36矩阵的下关节点位置，t是31载体平台。B. Dynamics Model The dynamics equations of the parallel manipulator are derived using Kan

19、e method, according to the theory, the active forces are equal to inertial forces, the dynamics state-space equation can be written by + +G （） = M （） + + V （, ） （3）B动力学模型并联机器人的动力学方程导出凯恩方法，根据理论，主动力等于惯性力，动态状态空间方程可以写的where M （） is the 66 mass matrix, V （, ） is an 61 vector of centrifugal and Coriolis t

20、erms, G （） is an 61vector of gravity terms, is a 61 vector of generalized applied forces.在M（）是66质量矩阵，V（，）是一个61向量的离心和科里奥利术语，G（）是一个61重力矢量，是61向量的广义力。The applied forces is transformed from mechanism actuator forces, which is given by = JlT *Fa （4）应用部队转化机制的执行力，这是由= JlT *Fa （4）where J l is a Jacobian 66 m

21、atrix of transformation between generalized velocity of platform and protraction velocity l of hydraulic cylinders, and Fa is a 61 vector representing cylinder forces.J1是雅可比矩阵66之间转换的广义速度平台和牵引速度的液压缸，Fa是一个61向量表示气缸的力量。The gravity term, G （） , contains gravitational constantg and generalized coordinate

22、, it depends only on , which can be described as G （） = G p + （Juc,ai*Jai） T mu .g+ （J dc,ai .J ai） T . m d .g） （5）重力，G（），包含引力常数G和广义坐标，只取决于它，可以描述为G （） = G p + （Juc,ai*Jai） T mu .g+ （J dc,ai .J ai） T . m d .g） （5）where Gp is upper platform gravity item, G p = m p *（g,0） T , mp is the total mass of up

23、per platform and load, and the 31 gravitational constant vector g = （0,0, g ） T , mu is the mass of piston, md is the mass of hydraulic cylinder, J uc,ai is a Jacobian 33 matrix of velocity transformation between upper joints and the piston center of mass, J dc,ai is a Jacobian 33 matrix of velocity

24、 transformation between generalized velocity and the hydraulic cylinder center of mass, and J ai, is is a Jacobian 36 matrix of velocity transformation between generalized velocity and upperjointsGp是上平台重力项，G p = m p *（g,0） T ,Mp是英国总质量上平台和负载，以及31重力常数r g = （0,0, g ） T ，Mu是质量是活塞，Md是液压缸的质量，Juc，ai是一个3雅可比

25、矩阵之间的转换3速度上和活塞质量中心，Jdc，ai是一个33的雅可比矩阵之间的速度变换广义速度和液压缸的质心，Jai，is 是是一个3雅可比矩阵之间的6速度变换广义速度和上接头CONTROL DESIGN控制设计In 6-DOF hydraulic driven parallel manipulator, PID controller is applied to achieve tracking control of platformextensively, which is called Joint Space （JS） control scheme.在六自由度液压驱动并联机器人，PID控制

26、可实现跟踪控制平台广泛，这是所谓的联合空间（JS）控制方案。The JS uses mechanism inverse kinematics for computing desired cylinder length trajectories from desired Cartesian trajectories, see Fig. 2.JS利用机构逆运动学计算所需的气缸长度轨迹所需的笛卡尔轨迹，见图2。 Figure 2. Joint space control scheme for 6-DOF hydraulic Parallel manipulator platform 图2。关节空间控

27、制方案的六自由度液压并联机器人平台The model-based controller considered the dynamic characteristic of parallel manipulator embedded the forward kinematics, dynamic gravity item and inverse of transfer of servovalve control hydraulic cylinders and inverse of transpose of Jacobian matrix （J l T ） 1 in inner control loop, see Fig. 3基于模型的控制器是动态特性的并联机器人的运动学嵌入，动