【机械类毕业论文中英文对照文献翻译】一个机器人结构设计及运动学
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英文原文THE STRUCTURE DESIGN AND KINEMATICS OF A ROBOTMANIPULATORml. THEORYKESHENG WANG and TERJE K . LIENProduction Engineering Laboratory, NTH-SINTEF, N-7034 Trondheim, NorwayA robot manipulator with six degrees of freedom can be separated into two parts: the arm with the first three joints for major positioning and the wrist with the last three joints for major orienting. If we consider theconsecutive links to be parallel or perpendicular, only 12 arm and two wrist configurations are potentially usefuland different for robot manipulator mechanical design. This kind of simplification can lead to a generalalgorithm of inverse kinematics for the corresponding configuration of different combinations of arm and wrist.The approaches for calculating the inverse kinematics of a robot manipulator are very efficient and easy.The approaches for calculating the inverse kinematics of a robot manipulator are very efficient and easy.1. INTROUCTIONA robot manipulator consists of a number of linksconnected together by joints. In robot manipulatordesign, the selection of the kinematic chain of therobot manipulator is one of the most importantdecisions in the mechanical and controller designprocess.In order to position and orient the end effector ofthe robot manipulator arbitrarily, six degrees offreedom are required: three degrees of freedom forposition and three degrees of freedom for orient-ation. Each manipulator joint can provide onedegree of freedom, and thus a manipulator musthave a minimum of six joints if it is to provide sixorthogonal degrees of freedom in position andorientation.The construction of manipulators depends on thedifferent combination of joints. The number of poss-ible variations of an industrial robot structure can bedetermined as follows:V =6where V= number of variations.D F = n u m b e r of degrees of freedomThese considerations show that a very largenumber of different chains can be built, for examplesix axis 46,656 chains are possible. 6 However, alarge number is not appropriate for kinematicreasons.We may divide the six degrees of freedom of arobot manipulator into two parts: the arm whichconsists of the first three joints and related links; andthe wrist which consists of the last three joints andrelated links. Then the variations of kinematic chainswill be tremendously reduced. Lien has developedthe constructions of arm and wrist, i.e. 20 differentconstructions for the arm and eight for the wrist.2In this paper, we abbreviate the 20 different armsinto 12 kinds of arms which are useful and different.We conclude that five kinds of arms and two kinds ofwrists are basic constructions for commercial indus-trial robot manipulators. This kind of simplificationmay lead to a general algorithm of inverse kinema-tics for the corresponding configuration of differentcombinations of arm and wrist. 2.STRUCTURE DESIGN OF ROBOT MANIPULATORSIn this paper, for optimum workspace and sim-plicity, we assume that:(a) A robot with six degrees of freedom may beseparated into two parts: the linkage consistingof the first three joints and related links is calledthe arm; the linkage of the remaining joints andrelated links is called the wrist.(b) Two links are connected by a lower pair joint.Only revolute and linear joints are used in robotmanipulators.(c) The axes of joints are either perpendicular orAccording to the authors knowledge, thisassumption is suitable for most commercially usedindustrial robot manipulators. We can consider thestructure of arm and wrist separately.2.1. The structure o f the arm o f robot manipulator(a) Graphical representation. To draw a robot inside view or in perspective is complicated and doesnot give a clear picture of how the various segmentsmove in relation to each other. To draw a robot in aplane sketched diagram is too simple and does notgive a clear construction picture. We compromisethis problem in a simple three-dimensional diagramto express the construction and movements of arobot manipulator. A typical form of representationfor different articulations is shown in Table 1.(b) Combination of joints. We use R to representa revolute joint and L to represent a linear joint.Different combinations of joints can be obtained asfollows:According to the different combinations with theparallel or perpendicular axes, each previous combin-ation has four kinds of sub-combination. Thus, 32combinations can be arrived at: If the second joint is a linear joint and both the otherjoints are perpendicular to it, two choices in relationto the first and the third joints are considered paral-lel or perpendicular.In all, there are 36 possible combinations of a simplethree-joint arm.Nine of 36 possible combinations degenerate intoone or two degrees of freedom.Seven of the remainder are planar mechanisms.Thus, there are 20 possible spatial simple arms.Let us consider R1 1 L2 I L3 in whichthe first joint permits rotation about the vertical axis,the second joint is a vertical linear joint (i.e. parallelto the first), and the third joint is a horizontal linearjoint (i.e. perpendicular to the second). This armdefines a typical cylindrical robot. Changing thesequential order of the joints so that either (a) thevertical linear joint precedes the rotary joint, or (b)the vertical linear joint follows the horizontal one,will result in no change in the motion of the arm. Inthis case there are two linkages which are bothequivalent to the standard cylindrical linkage. Inall such cases where two or more equivalent linkagesexist, the representative of the group will be the onein which the linear joint that is parallel to a rotaryjoint is in the middle (joint No. 2). Counting onlyone linkage to represent the group of equivalentswill eliminate eight of the 20 combinations. Theremaining 12 categories of links are useful and dif-ferent shown in Fig. 1. We get the same results as inRef. 4.(c) Five basic types o f manipulator arm. Althoughthere are 12 useful and different arm-configurationswhich can be used in the design of a robot man-ipulator arm, in practice only some of them arepractical and commonly used. We find that mostcommercially available industrial robots can bebroken down into only five groups according to the.characteristics of their arm motion and geometricalappearance.The five groups can be defined as follows and areshown in Fig. 6.1. Cartesian ( L I L I L)2. Cylindrical (R II L 1 L)3. Spherical (R I R I L)4. Revolute (R I RII R)5. Double cylindrical ( LII R II R).2.2. The structure o f a manipulator wrist(a) Joint type. We have used the first three joints,i.e. the arm of the robot manipulator, to completethe major task of positioning. Then we use the lastthree joints to provide the three degrees of freedomof orientation and refer to the related linkages as thewrist.The wrist of a complete manipulator must containthree revolute joints, since the orientation of a rigidbody has three degrees of freedom, for example firstrotation about the X axis, then rotation about the yaxis, and finally rotation about the z axis.(b) Combination or joints and links. Because theorientation of a wrist which only has three rotationaljoints is simplest, its combination is much simpFrom the combination R R R , we know that onlyone of the four configurations can be used for com-pleting the orientation of robot wrist. R II R II R is aplanar mechanism. R 1 R II R and R II R 1 R cannotexpress three degrees of freedom in the orientationof the robot wrist. So only the R 1 R 1 R construc-tion can be used to complete the orientation task.If we have a different sequence of x, y, z axes, ofcourse we can get many kinds of wrist configuration.But many of them are equivalent. We only con-sider the relationship between the first and the thirdjoint: parallel and perpendicular. Two differentcombinations can be arrived at, i.e. the Euler angleand r o l l - p i t c h - y a w angle expressions that are shownin Fig. 2. The sequence of x, y, z axes does, however,have an influence on the complexity of the inversekinematic solution.2.3. Typical robot manipulator structure We can use five categories of arm configurationand two kinds of wrist configuration to combine 10different kinds of robot manipulators with the sixdegrees of freedom which exist in industrial practice.Of course, we can also consider the other seven outof 12 arm categories with one out of two wristcategories to build a new robot manipulator. Butmost of them have not appeared in industrial prac-tice yet.3. SOLUTION FOR INVERSE KINEMATICS OF ROBOT MANIPULATOR3.1. General principlesTo find the inverse kinematic equations of a robotmanipulator at first appears to be a difficult task. Butwhen the manipulator is separated into two parts, itbecomes relatively simple.The relationship between the position and orien-tation of manipulator links connected together byrotational joints shown in Fig. 3, can be described byWhere0i is the ith joint variable;di is the ith joint offset;ai is the ith link length; andai is the ith link twist angle.The position and orientation of the end effector ofthe robot manipulator T is the matrices product. 3,T = A I A 2 A 3 A 4 A s A 6 . (2)By the associative law the product of matrices can beregrouped into two subsets which represent the armand wrist respectivelyWhereAndThe superscripts designate the reference frame; arepresents the tip of the arm; and w represents thetip of wrist, i.e. the center of the end effector of themanipulator.T given for the end effector can be written as a4 x 4 homogeneous matrix composed of a orienta-tion submatrix R and a position vector p5.6We can obtain the vector OaPdirectly using a vectoranalysis method. The detail will be mentioned in thenext section.from Eq. (4),We can get 01, 02, 03, the first three joint variablesfrom the solution of the following equation:The orientation of the end effector of the robotmanipulator can be considered as the product of theorientation of the arm and the orientation of the wrist: From Eqs (12) and (5), we can obtain where We can get the last three joint variables 04, 05, 06 by solving Eq. (13).3.2. Different methodsThere are two kinds of solutions for the robotmanipulator: closed form solutions and numericalsolutions. Because of their iterative nature, numeri-cal solutions are generally much slower than thecorresponding closed form solutions, so much so that for most uses, we are not interested in the numerical approach to solution of kinematics. But, in general, it is much easier to obtain the numerical algorithmthan to obtain the closed form solution.In this paper we propose algorithms of both solu-tions.(a) Closed form solution. In the closed form solu-tion, the key problem is to obtain the position of thetip of the arm P. It is simple to obtain the position ofthe arm tip for the wrist axis intersecting at onepoint. But it is complex for the wrists where there isan axis offset, because the movement of the wristwill greatly affect the position of end effector of themanipulatorIn the following, we use the RRR + Euler angleand RRR + R - P - Y angle as examples to describehow to get the position of the tip of arm separately. RRR + Euler angleFigure 4 shows a sketch diagram of a R R R + Euler angle robot manipulator (PUMA 600) and the co-ordinate system which is represented by the D - Hexpression. The figure shows the relationship be-tween the arm and wrist vectors. r, is the positionvector from the base coordinate frame to the centerof the end effector of the robot manipulator. Arepresents the approach direction of the end effec-tor, aPis the arm vector measured from the origin tothe connecting point of the arm and wrist, gP is thewrist vector having the same direction as the Avector and length measured from the connectionpoint of the arm and wrist to the center of the endeffector.With reference to frame 0, the product R gP issimply gP, i.e. the position of the center of the endeffector of robot manipulator measured from the tipof the arm, all with respect to frame 0. We canobtainThis states that the total translation of the endeffector is the sum of the translation from the base tothe tip of the arm plus the transformation from thetip of the arm to the center of the end effector.From Eq. (17), we can easily obtain the positionof the arm tip P as follows:Then we can use Eqs (10) and (11) to obtain the firstthree joint variables 0:, 02, 03 and Eq. (13) to obtainthe last three joint variables 04, 05,06. The detailedsolution is shown in Part II. t0Figure 5 shows a sketch diagram of a RRR +R - P - Y angle robot manipulator (Cincinatti Mila-cran T 3) and the coordinate system. Euler anglesare different from R - P - Y angles because the vector0p is affected by the movement of joint 4. Here is anexample showing how to treat the wrist axis offset.gPt:is the wrist vector having the same direction asthe A vector and length measured from the point ofjoint 4 to the center of the end effector, i.e. d+. P2 isthe other wrist vector having length measured frompoint of joint 4 to point of joint 5, i.e. a4. oP, theposition of arm, can be computed from the se-quential solution of the following set of equations:Then we can obtain 01, 02, 03 from Eqs (10) and (11)and obtain 0+, 05, 06 from Eq. (13). General closed form solution algorithmStep 1. Finding the approach vector of the endeffectorStep 2.If there is some off-set in the wrist construc-tion, use the vector algebra to determine theoff-set gP, and get the arm vector, i.e. theposition of arm tip, then go to step 4.Otherwise go to Step 3. Compute the arm vector P directly usingapproach vector A.Step 4. Compute the first three joint variables 01,02, 03, using the arm vector gP from Eqs(10) and (11).Step 5. Compute the last three joint variables 04, 05,06 from Eq. (13).This approach shows that the number of computa-tions is kept to a minimum by reducing the overallproblem into separate steps which in turn lowers thelikelihood of errors and helps to reduce the tedious-ness of the work.(b) Numerical solution. The algorithm for thenumerical solution:Step 1. Assume the last three joint variables 04, 05,06 by the best available approximation,perhaps from a previous computed point.Step 2. Compute the arm joint variables 81, 02, 03from Eqs (10) and (11).Step 3. Compute wrist joint variables 04, 05, 06 fromEq. (13), using the values of the arm jointvariables obtained from step 2.Step 4. Compute the position and orientation of theend effector of robot manipulator using thevalues of all joint variables obtained fromstep 2 and step 3.Step 5. If the errors between the given values andthe calculated values is less than a pre-specified value, then the procedure stops.Otherwise go to step 2 to repeat the pro-cedure.The physical interpretation of the above pro-cedure is alternately to move the arm and wrist, oneto satisfy the positional and other to satisfy theorientational specification of the end effector, eachtime moving only the arm (or the wrist) while hold-ing the wrist (or the arm) fixed.This method has been implemented in a PUMA600 robot manipulator. It has been found that four is a sufficient number of iterations to reach therequired accuracy (A 0.01 mm) and the number has been fixed in the inverse kinematic solution.This algorithm has the advantage of treating the different kinds of robots with the same algorithm.But this method needs so much more computing time than the closed form solution, that it is notsuitable for real-time control of robot manipulators.4. CONCLUSIONSThe variety of possible robot configurations isvery large. A step towards generalization has been made by emphasizing that robot manipulators ofpractical importance are separable into primary sub-systems, the arm and the wrist. Mathematical treat-ment of various robots may be modularized and thusgreatly simplified by giving a separate description ofvarious arms and various wrists in common use.It has been discovered that only 12 useful and different categories of arm construction and twokinds of wrist construction exist. Using thehomogeneous transformation matrix method, theinverse kinematic solution is easily derived.The two algorithms which consist of the closedform and numerical solution of the inverse kine-matics have been given in this paper.REFERENCES1. Denavit, J., Hartenberg, R.S.: A kinematic notationfor law pair mechanisms based on matrices. J. Appl.Mech. Trans. ASME 77: 215-221, 1955.2. Lien, T.K.: Banestyring for universelle handterings-automater. Trondheim, August 1980.3. Lien, T.K.: Coordinate transformations in CNC sys-tem for automatic handling machines, llth CIRPSeminar on Manufacturing Systems, Nancy, France,June 1979.4. Milenkovic,V., Huang, B.: Kinematicsof major robotlinkage. 13th International Symposium on IndustrialRobots and Robotics 7, Vol. 2, pp. 31-46, 1983.5. Paul, R.P.: Robot Manipulators: Mathematics, Pro-gramming, and Control. MIT Press, Cambridge,1982.6. Lee,. C.G.S.: Fundamentals of Robotics. Addison-Wesley, New York, 1983.7. Warnecke, H.J., Schraft, R.D.: Industrial Robots. IFS,Bedford, 1982.8. Pieper, D.L.: The kinematics of manipulators undercomputer control. AIM 72, Stanford, CA. StanfordUniversity Artificial Intelligence Laboratory.9. Coiffet, P., Chirouze, M.: An Introduction to RobotTechnology. Kogan Page, London, 1983.10. Wang, K., Lien T.K.: Closed form solution for theinverse kinematics of a PUMA robot man-ipulator-II. Demonstration. Robotics Comput.-Integr. Mfg. 5: 159-163, 1989.中文原文一个机器人结构设计及运动学机械臂毫升.理论KESHENG WANG and TERJE K . LIEN生产工程实验室，NTH-SINTEF，N-7034，挪威特隆赫姆 六自由度机器人可以分为两个部分：与前三个关节为主要定位，最后三个关节为主要面向腕臂。如果我们考虑连续的链接是平行或垂直的，只有12的臂和两个手腕结构可能是有用的而且不同于对机器人机械手的机械设计。这种简化可以导致对手臂和手腕的不同组合配置相应的逆运动学算法。对于一个机器人逆运动学是非常有效和简单的计算方法。 简介 一个机器人由若干环节通过接头连接在一起。在机器人的机械手设计，对运动链的选择是器人一个最重要的决定在机械和控制器的设计过程。 为了定位和定向的机器人末端执行器的任意，六自由度的要求：方向三度的位置和三自由度的自由。每个机械手关节可以提供一个自由度的机械手，因此必须要提供在六个自由度的位置和方向正交的至少有六的接头。 机械手的结构取决于节点的不同组合。对工业机器人的结构的可能变化的数量可以确定如下。V=6DF；那么V=数量的变化；DF=自由度 这些因素表明，不同链可建数量非常大，例如六轴46656链是可能的。然而，这是大量不适合运动的原因。 我们可以将一个机器人六自由度分为两部分：臂由前三个关节和相关链接;与手腕由过去的三节点和相关链接。然后运动链的变化将极大地减少。即留置了手臂和手腕的结构。20种不同的手臂和8种手腕设计。 在文本中，我们有20种不同的手臂，有12中手臂是不同的，很有用的。我们得出这样的结论：五种手臂和两种手腕是商业工业机器人的基本结构。这种简化可能导致对手臂和手腕的不同组合的相应配置逆运动学算法。机器人的结构设计机械手 在本文中，最佳的工作空间和简单，我们假设：（一）具有六个自由度的机器人可以分为两部分：连接组成的前三个关节和相关的链接被称为ARM；剩余的关节联动相关链接是所谓的手腕。（二）两个环节由一个联合低副连接。旋转和直线连接中使用的机器人机械手。（三）接头的轴线是垂直或相互平行。 据作者所知，这种假设是适用于大多数的商业工业机器人。我们可以考虑结构的手臂和手腕的分别。对机器人的手臂结构（1） 图形表示。画一个机器人在侧视图或透视是复杂的和不放弃的各个环节中的相互关系，如何清晰的照片。在一个平面上画一个机器人绘图过于简单，并没有给出一个明确的施工图。我们妥协的这个问题的一个简单的三维图表示的机器人机械手的结构和动作。对不同关节表示的一种典型形式显示在表：表1 一个机器人的图形表示 类型 运动 自由度 符号1. 固定梁 固定 0 2. 转动 旋转 1 3 线性 翻译 1 1 2 3 4 5 6 7 8RRR RRL RLR RLL LRR LRL LLR LLL（二）相结合的关节。我们使用R来表示一个转动关节和L代表一个线性联合。接头不同的组合可以得到如下： 根据与平行或垂直的轴的不同组合，每一组合有四种亚相结合。因此，32的组合可以到达：(1) RRR RRR (2)RRL RRL RRR RRL RRR RRL RRR RRL(3) RLR RLR (4)RLL RLL RLR RLL RLR RLL RLR RLL(4) LRR LRR (5)LRL LRL LRR LRL LRR LRL LRR LRL(6) RRR LLR (7)RRL LLL LLR LLL LLR LLL LLR LLL如果第二节是一种线性联合和其他关节垂直于它，与第一、第三节，两个选择被认为是平行或垂直在所有的，有36种可能的一个简单的三节臂组合。 36种的9种可能的组合退化为一个或两个自由度。其余的七种都是平面机构，因此，有20种可能的空间简单的手臂。让我们考虑 我绕垂直轴旋转的第一关节允许，第二节是一个垂直的线性联合（即平行到第一）。而第三个关节是一个水平，联合（即垂直于第二）。该臂定义了一个典型的圆柱机器人。改变关节的顺序，无论是（一）垂直线性联合之前的旋转接头，或（二）垂直线性联合为一个水平，将导致在手臂的运动没有变化。在这种情况下，有两个机构都是“等效”标准圆柱联动。在所有这些情况下，两个或两个以上的等效机构存在，本集团的代表将一其中线性连接在平行于旋转关节（关节中2号）。只计算一个连锁代表等同组将消除20个组合的八个。其余的12个类别的链接是有用的，不同的图1所示。我们得到的结果是一样的参考4。（三）五种基本类型的机械臂。虽然有12个有用的和不同的臂结构它可以用在一个机器人机械手的设计臂，在实践中，只有其中的一些实用和常用的。我们发现，最市售的工业机器人可以分成五组根据一个机器人结构设计及运动学。图1 手臂配置十二种类型。他们的手臂运动的外观和几何特性。五组可以定义如下，图6所示。1.笛卡尔（LLL）2.圆柱（RLL）3.球形（RR L）4.旋转（RRR）5.双柱(LLR)一个机械手的手腕结构（一）联合型。我们使用的前三个关节，即机器人的机械臂，完成定位的主要任务。然后我们用最后的三节提供的三自由度的方向和参考相关联系的手腕。一个完整的机械手的手腕必须包含三个旋转接头，由于刚性取向身体具有三个自由度，例如第一关于X轴旋转，然后绕y轴，最后绕Z轴.（二）的组合或关节和链接。因为一个手腕，只有三的旋转方向关节是最简单的组合，它是非常简单的。，它是一个组合比手臂更简单。从组合的存款准备金率，我们知道只有四种配置可用于完成机器人手腕的方向。是一个平面机构。和不能在三个自由度的定位表达机器人的手腕。所以只有RRR建设可以用来完成定位任务。 如果我们有X，Y，Z轴的顺序不同，当然我们可以得到多种手腕配置。但他们中的许多人是“等效”。我们只考虑第一和第三的关系关节：平行和垂直。两个不同的组合可以到达，即欧拉角和辊间距横摆角的表达式，证明图2。X，Y，Z轴次序，确实，但是，对逆复杂性的影响运动学求解。图2 两种类型的手腕配置。 器人的典型结构 我们可以用五种臂形两种手腕配置结合10不同种类的六个机器人在工业实践中存在的自由度。当然，我们也可以考虑其他的七12臂类二分之一的手腕类来建立一个新的机器人。但他们中的大多数都不在工业实践中出现。3.逆运动学求解机器人3.1.一般原则找到一个机器人逆运动学方程机械手首先似乎是一项艰巨的任务。但当机器人分为两部分，它变得相对简单。位置和方向之间的关系机械手的链接连接在一起的旋转接头图3所示，可以描述迪纳维特哈坦伯格矩阵。图3 D-H坐标系参数。如下： 机器人操作臂的T末端执行器的位置和方向的矩阵乘积。通过联想法矩阵的产品可以重组为两个亚群为代表的手臂和手腕的分别。这儿和上标指定参考帧；一个代表的臂的前端和W代表；尖端的手腕，即对末端执行器的中心机械手。T为端部执行器可以写成44齐次矩阵由一个方向子矩阵R和位置矢量p5.6。我们可以得到向量的直接使用矢量分析方法。细节将在下一节中提到。从4式我们可以得到三个关节变量从下面的方程的解：机器人末端执行器的方向机械手可以看作是该产品的臂的取向手腕：从式（12）和（5），我们可以得到：然后我们可以得到最后三个关节变量04，05，06 求解方程（13）。 不同的方法 有两种解决方案的机器人机械手：封闭形式解和数值解决方案。因为他们的迭代性质，数值解决方案通常比慢得多相应的封闭形式的解决方案，所以在大多数应用中，我们不感兴趣的数值以运动学求解方法。但是，在一般情况下，这是很容易获得的数值算法比获得的封闭形式的解决方案。有两种解决方案的机器人在本文中我们提出了两个解决方案的算法。封闭形式解。在封闭的形式解决方案，问题的关键是要获得的位置臂的前端P.它简单得位置对腕关节轴相交于一个臂的前端点。但它是复杂的手腕那里轴线偏移，因为手腕的运动将极大地影响的末端位置机械手。在下面，我们使用RRR+欧拉角与RRR + r-p-y角作为例子来描述如何获得的机械臂末端的位置。RRR+欧拉角:图4显示一个RRR+欧拉示意图角机器人（彪马600）和坐标系统采用D-H表示表达。图中显示的关系向量的手臂和手腕。 R，是位置从基坐标系中心向量机器人操作臂的端部执行器。一个代表了末端执行器的方向，AP是从起源到测量臂载体的手臂和手腕的连接点，GP是具有相同的方向矢量的手腕从连接测量矢量和长度的手臂和手腕点结束的中心效应。参考框架0，产品简单的，即结束的中心位置机器人从尖端测量效应的手臂，所有相对于框架0。我们可以获得这说明，最后的总翻译效应是从翻译的总和的手臂加上从转化的尖端臂的前端到末端执行器的中心。从式（17），我们可以
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