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六自由度剪式并联机床结构设计,自由度,并联,机床,结构设计
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LE系列电感式接近开关产品外观图产品尺寸图点击图片放大显示 LE系列型号说明 输出形式型号接线方式直流NPN三线常开LE4-1K见（1）号接线图直流NPN三线常闭LE4-1B直流PNP三线常开LE4-2K见（2）号接线图直流PNP三线常闭LE4-2B直流二线常开LE4-5K见（5）号接线图直流二线常闭LE4-5B直流NPN四线常开+常闭LE4-3见（3）号接线图直流PNP四线常开+常闭LE4-4见（4）号接线图交流二线常开LE4-6K见（6）号接线图交流二线常闭LE4-6B LE系列参数说明 安装形式非埋入式工作电压直流型：10-30VDC检测距离4mm20%静态电流DC三线型2.5mA设定距离03.2mm响应频率300Hz回差值小于检测距离的10%电流输出200mA标准检测体15151铁防护等级IP66残留电压DC三线式：1.5VDC，DC二线式：4VDC，AC二线式：8VAC温度影响在-25到+70范围内，对在+25时的检测距离是在10%以下绝缘电阻50M Min （500VDC Mega基准）耐压1500VAC 50/60Hz 一分钟抗振动抗振动：10-55Hz（周期每分钟）复振幅1mm，X、Y、Z各方向2小时抗冲击抗冲击：500M/S2( 50G ) X、Y、Z各方向3次环境温度工作时 : -30 to +80(未结冰状态下)储存时 : -30 to +80未结冰状态下 )环境湿度工作时: 35 to 95% RH指示灯动作显示（红色LED） 输出接线图 安装形式非埋入式工作电压直流型：10-30VDC检测距离4mm20%静态电流DC三线型2.5mA设定距离03.2mm响应频率300Hz回差值小于检测距离的10%电流输出200mA标准检测体15151铁防护等级IP66残留电压DC三线式：1.5VDC，DC二线式：4VDC，AC二线式：8VAC温度影响在-25到+70范围内，对在+25时的检测距离是在10%以下绝缘电阻50M Min （500VDC Mega基准）耐压1500VAC 50/60Hz 一分钟抗振动抗振动：10-55Hz（周期每分钟）复振幅1mm，X、Y、Z各方向2小时抗冲击抗冲击：500M/S2( 50G ) X、Y、Z各方向3次环境温度工作时 : -30 to +80(未结冰状态下)储存时 : -30 to +80未结冰状态下 )环境湿度工作时: 35 to 95% RH指示灯动作显示（红色LED） Manipulability Analysis of a Parallel Machine Tool: Application to Optimal Link Length Design Keum-Shik HongSchool of Mechanical EngineeringPusan National University. 30 Changjeon-dongKeumjeong-ku, Pusan, 609-735, Koreae-mail: kshonghyowon.pusan.ac.krJeom-Goo KimGraduate CollegePusan National UniversityKeumjeong-ku, Pusan, 609-735, Koreae-mail: jgkim92hyowon.pusan.ac.krReceived November 5, 1999; accepted March 23, 2000In this article, an input-output transmissivity analysis of the EclipseTM, which is a parallel machine tool capable of five-face rapid machining, is investigated. By splitting the weighted Jacobian matrices into two parts, the linear velocity, angular velocity,and force-moment transmissivities are analyzed. A new manipulability measure,which combines the volumes of manipulability ellipsoids and the condition numbersof the splitted Jacobian matrices, is proposed. Two link parameters, the radius of theupper platform and the length of a supporting links, are optimally designed bymaximizing the new manipulability measure introduced. Computer simulations areprovided. 2000 John Wiley & Sons, Inc.1. INTRODUCTIONAs the time-to-market of new products becomes shorter, rapid machining becomes more important to gain competitiveness, where rapid means the extreme reduction of the machining lead-time re-Correspondence to: Keum-Shik Hong.quired to complete a part from the original work- piece. Rapid machining is especially required when manufacturing various prototypes of the final prod- uct. To minimize the machining lead-time, setup changes should be avoided. Once the original work- piece is set up on a machine tool, the machine tool should execute all the necessary machining pro- cesses to machine all five faces of the workpieceJournal of Robotic Systems 17(8), 403-415 (2000) 2000 by John Wiley & Sons, Inc.Hong and Kim: Investigating the Analysis of Eclipse415without any setup changes. Here, the five faces mean the top and four lateral surfaces of a prismatic workpiece.Conventional computerized numerical controlYoshikawa5 gave one of the first comprehensive mathematical treatments for the manipulability of general open chains. Gosselin and Angeles6 and Zanganeh and Angeles7 proposed a number of coor-CNC.machining centers are generally classifieddinate-based formulations for the kinematic man-into vertical and horizontal types according to thedirection of the spindle axis. Since the spindle axisis fixed in most machining centers, the setup of aprismatic workpiece must be changed more thanonce to process all five faces. Even though five-facemachining plano-millers are used, they are excep-tional machines for very large workpieces such asmachine tool beds and columns.Since the conventional machining centers arevery inefficient for five-face machining with a singlesetup, the study of a new machine tool structurewas performed to realize rapid machining purpos- es.1 - 3 Several machine tool manufacturers have de- veloped commercial machining centers based on the parallel platform architecture: For example, Gidding& Lewis, Ingersoll, Geodetics, and Hitachi Seiki, etc.However, none of the parallel machine tools canperform five-face rapid machining. This is becausethe maximum tilting angle of the spindle from the vertical posture is 300 or less. However, the EclipseTMpresented in this article is designed to have a large workspace over the other parallel machine tools. The spindle can move continuously from the verti- cal direction to the horizontal orientation, and canrotate 3600 about the Z axis. Because of these ad-vantages, the Eclipse can perform five-face rapidmachining and several successive machining pro-ipulability of parallel manipulators. Also, Kosugeet al.8 analyzed, by splitting a Jacobian matrix intotwo parts, the output force and moment transmis-sion to the upper platform for parallel manipulators.Ahn and Hong9 analyzed the direction of the inputforce that maximizes or minimizes the output forceand moment through the manipulability ellipsoidbased on the singular value decomposition of asplitted Jacobian matrix. Kokkinis and Paden10 pre-sented the force and velocity polytopes instead ofthe manipulability ellipsoid, and then combinedthem with geometric operations. Park and Kim11presented a different geometric formulation of themanipulability for general closed chains. Chiacchioet al.12 analyzed the force polytope and force ellip- soid for redundant manipulators.In this article, the inverse and forward kinemat- ics, velocity and angular velocity transmissivity, force and moment transmissivity, and manipulabil- ity of the Eclipse, which is a new type of parallel machine tool,1 are investigated. By extending thework of Kosuge et al.,8 the input-output transmis-sivity analysis is performed by splitting Jacobianmatrices into two parts. In ref. 8, the range of outputforce was estimated by restricting output momentat zero, while the range of output moment wasestimated by restricting output force at zero. Thiscessesturning, milling, and grinding.within thesituation, however, is not general because both forcesame platform. Currently, a second-generation pro-totype of the Eclipse has been introduced by im-proving the first prototype in the areas of pathgeneration and elimination of kinematic singulari-ties.3This article analyzes how the input forces andvelocities of actuated joints of the Eclipse are trans- mitted to the upper platform. The input-outputtransmissivity of a mechanism is intimately related with the manipulability of the mechanism. There- fore, the manipulability of the Eclipse is analyzed with the introduction of a new manipulability mea- sure. Then, two link parameters, the radius of the upper platform and the length of a supporting links, are designed by maximizing the new manipulability measure introduced.The studies related with the manipulability of mechanisms are: Salisbury and Craig4 first used the notion of manipulability to design robotic fingers.and moment are simultaneously transmitted to theupper platform. Therefore, this article analyzes the range of the force-moment of the upper platformwithout assuming that either the output force or the output moment is in the null space. This analysis is based on the singular value decomposition and the pseudo inverses of the splitted Jacobian matrices. In addition, a new manipulability measure, which is composed of the condition number of the splitted Jacobian matrix and the total volume of the manipu- lability ellipsoid, is proposed. Then, by maximizing this new manipulability measure, optimal link lengths are designed.The contributions of this article are: First, this article is the first article dealing with the input-out-put transmissivity and manipulability analysis for the Eclipse. Second, a new manipulability measure, which includes the total volume of manipulability ellipsoids and the condition numbers of Jacobianmatrices, is introduced. Third, two link parameters, the radius of the upper platform and the length of a supporting links, are designed by maximizing the manipulability measure defined.This article has the following structure: Section 2 discusses the kinematics of the Eclipse. The for- ward kinematics, inverse kinematics, and workspace analysis are also performed. In Section 3, Jacobianmatrices and force-moment transmission relation-ships of the Eclipse are derived. The characteristics of the input-output force and velocity of the Eclipseare analyzed in Section 4. A number of manipulabil- ity measures are proposed in Section 5. In Section 6, the analyzed results are applied to design optimal link lengths of the Eclipse. Finally, conclusions are presented in Section 7.sliding on the circular ring of the lower platform. There are also three supporting links between three vertical columns and the upper platform. The radius of the upper platform is r and that of the lower platform is R. Also, the joint between a vertical column and a supporting link is a 1-DOF revolutejoint Ci i = 1, 2, 3. The joint between a supportinglink and the upper platform is a 3-DOF ball joint Bii = 1, 2, 3. The 6-DOF motion of the upper platformis obtained by six actuated prismatic joints, three onthe circular ring and three on the vertical columns,as indicated in Figure 2. The Eclipse, however, hastwo additional actuators, indicated by C1 and C2 , to overcome singular configurations.13,3 It is also noted that one out of three sliders should be located below the upper platform to avoid a possible interferencebetween the spindle motor attached to the upper platform and a supporting link when the upper2. THE ECLIPSE: KINEMATIC ANALYSISplatform is tilted about 900. In Figure 2, C1is lo-Figure 1 is the first prototype of the Eclipse. Figurecated below the upper platform, while C2are located above the upper platform.and C32 is a simplified model for the kinematic analysis of the Eclipse. The Eclipse is a parallel mechanism because the upper platform is connected in parallel with the lower platform by three vertical columnsIn Figure 2, the F4 frame using X-Y-Z coordi-nates denotes the base frame whose origin is locatedat the center of the lower platform. The moving frame M 4 in x -y -z coordinates is affixed to theupper platform. Note thatdenotes the vectors represented in the moving frame. Let the origin ofthe upper platform be oc = Xc YcZc T. The rota-tion matrix of the upper platform is given by Z-Y-Zfixed angles , 户, . asn xR = n ynzsxaxsya yszaz= RotZ .RotY 户 .RotZ . 1.Figure 1. The first prototype of the Eclipse.Figure 2. A kinematic model of the Eclipse.where , 户, and are the rotating, tilting, andyawing angles of the upper platform, respectively.to bi. Hence, ei are determined asLet R = n s arepresent the orthogonal basis ofei = arctan 2 bi y , bi x .i = 1, 2, 34. M 4. Also, by combining the rotation matrix R withthe position vector Xc YcZc T, the following ho-where arctan 2 denotes the two argument arctangentmogeneous transformation matrix T is defined in ref. 14, p. 30. Then,function such that arctan 21, -1. = 3rj4.Second, the prismatic joint variables dii =n xT = n ysxaxsya yXcYc2.1, 2, 3. of three sliders are obtained. Let the length ofa supporting link be l. Then, the following con-straint equation holds:nzszazZcl = OCi - Ooc - oc Bii = 1, 2, 35.0001From 5., the following three scalar equations hold:Now the following notation is introduced.O, oc : The origins of the lower and upper plat-ii iiiwhered 2 - 2 d + 2 - 已 = 0i = 1, 2, 36.forms, respectively.ei , i = 1, 2, 3: The angular displacements fromthe X axis to joints Ai , i = 1, 2, 3.di , i = 1, 2, 3: The translational displacements ofprismatic joints Ci , i = 1, 2, 3.i = n z bix + az bi z + sz biy iiy i xy i zy i yc已 = l 2 - R sin e - n b - a b - s b - Y .2ix ixx i zx iyc- R cos e - n b - a b - s b - X .2i , i = 1, 2, 3: The angles made by three sup-Two roots of6. are: i + 已iand i - 已i . Thisporting links and the horizontal plane atindicates that there exist two possible solutions forCi , i = 1, 2, 3.each difor a given position and orientation of theThe DOF of the Eclipse is obtained using theDOF equation for spatial motion mechanisms15 asupper platform. However, d1 should be the smaller one, i.e., 1 - 已 1 , which prevents a possible colli-sion between the spindle motor and a supportingjlink by lowering C1below the upper platform.F = l - j - 1. + fi = 611 - 12 - 1. + 18 = 6i=13.Therefore, it can be said that the inverse kinematic solution is unique. That is, d1 = 1 - 已 1 , d2 = 2+ 已 2 , and d3 = 3 + 已 3 . It is also noted that 已iin 6. must be positive because已iis a part ofwhere F is the DOF of a mechanism, is the DOF of the space within which the mechanism operates, l is the number of links, j is the number of joints, andfi is the DOF of the ith joint. For the Eclipse, = 6,l = 11, and j = 12.solution. This can always be achieved by properlychoosing parameters R, r, and l.2.2. Forward KinematicsThe forward kinematic problem is to determine the position and orientation of the upper platform by2.1. Inverse Kinematicsusing six actuated joint variables, eiand dii =The inverse kinematic problem is to find the actu-1, 2, 3. As is common with most parallel mecha-nisms, it is very complicated to solve the forwardated joint variables, which are eii = 1, 2, 3. and dikinematics analytically because the values of i arei = 1, 2, 3., for a given position and orientation ofthe upper platform. First, the angular displacements ei of three vertical columns are derived. In Figure 2, the position vectors to the ball joints bi i = 1, 2, 3. in M 4 are fixed values. Therefore, the vectors bi i =1, 2, 3. transformed to F4 are calculated by premul-tiplying the homogeneous transformation matrix Tnot readily known.16 - 18 If all s are known, theiforward kinematics problem of the Eclipse can beeasily decoupled into three forward kinematic prob-lems for three serial subchains.Let each subchain be a Rei . - P di . - Ri .serial mechanism, where R and P denote a revolutejoint and a prismatic joint, respectively. Then, withTable I. D-H parameters of three subchains i = 1, 2, 3.Jointseda0. The position vector of the origin of the upper platform is now determined from 7. asTb + b + b1ei0R0oc =XcYcZc=123310.21800di0903i0l0Also, the orientation of the upper platform is deter- mined as19the Denavit-Hartenberg parameters defined inn = b1 - b211.Table I, the position vectors biare derived, using1b1 - b2 1the joint variables ei , di , and i , asb1 + b2 .j2 - b3s = .12.bi =bix biy =biz R cos ei - l cos ei cos iR sin ei - l cos i sin eil sin i + di7.b1 + b2a = n X s2.3. Workspace Analysisj2 - b313.The passive joint variables, i , which denote theangles between supporting links and the horizontalplane, can be determined by using the followingconstraint equation: 22In robotics literature, the workspace is often classi- fied into two parts, a reachable workspace and a dexterous workspace: The reachable workspace is the entire set of points reachable by the end-effector, whereas the dexterous workspace consists of thoseBi Bj= OBj - OBii = 1, 2, 3 j = 2, 3, 18.points that the end-effector can reach with an ar- bitrary orientation.20 However, these definitionsThe evaluation ofscalar equations,8. yields the following threeare not suitable for the Eclipse, because the Eclipse is a machining tool. The following definition of workspace for the Eclipse is proposed.2,3f1 1 , 2 . = C11 sin 1 sin 2 + C12 cos 1 cos 2+ C13 sin 1 + C14 sin 2Workspace: The total Cartesian workspace of the Eclipse is defined as the set of all achievable posi- tions that can be reached by the spindle tip for the+C15 cos 1 + C16 cos 2 + C17 = 0orientation set9004. , 户, . 1 00 豆 户 豆 900, -900 豆 豆f2 2 , 3 . = C21 sin 2 sin 3 + C22 cos 2 cos 3+ C23 sin 2 + C24 sin 3+C25 cos 2 + C26 cos 3 + C27 = 0f3 3 , 1 . = C31 sin 3 sin 1 + C32 cos 3 cos 1+ C33 sin 3 + C34 sin 1+C35 cos 3 + C36 cos 1 + C37 = 09.Note that there is no restriction on the rotating angle, , of the upper platform. Only the tilting andyawing angle of the spindle are restricted as above. To calculate the actual workspace, the following physical constraints should be further considered: i. The stroke limit of the prismatic joints on the vertical columns; ii. The minimum angle between two vertical columns; iii. The limit of the revolutejoint angles;iv.The limit of the spherical jointwhere coefficients Cij i = 1, 2, 3, j = 1, 2, 3, 4, 5, 6, 7. are functions of ei and di. The detailed expressionsof Cij are skipped considering the length of this article. For 9., two methods can be applied to solvei i = 1, 2, 3. The first method is to use an extra sensor, that is, to measure one angle out of 1, 2 , and 3. If one angle is measured, a closed formsolution to 9. can be derived. The second method isa numerical analysis. Detailed discussions on theseissues are skipped.angles. For the Eclipse, the actual values of thesephysical constraints are listed in Table II.3. JACOBIAN: SINGULARITY AND FORCE TRANSMISSIVITYIn this section, the Jacobian matrix, which repre- sents the relationship between the actuator veloci- ties and the velocity vector of the upper platform, isTable II. Kinematic constraints of the Eclipse.-44.25 豆 d1 豆 0.0where Jre R3X 6, Jr R3X 3, and subscript r repre- sents the rotational angular velocity component. Now, by differentiating constraint equation 9., theStroke limits cm9.75 豆 d2 豆 45.759.74 豆 d3 豆 45.75following relationship between the active joints andthe passive joints is obtained. Then,Minimum angle between300two vertical columnsRevolute joint limit angle-900 豆 豆 900Spherical joint limit angle600。 F。 F 9 + 全 Fe 9 + F = 0。 9。 17.where F = f1 f2f3 T,Fe = 。 Fj。 9, andF =first of all derived. Then, using this Jacobian matrix, singularities and force-moment transmission rela-tionship are further analyzed. The Jacobian matrix is obtained by differentiating the forward kinematic。 Fj。 . By substituting 17. into 15. and 16., therelationship between the active joint velocity vectorand the velocity vector of the upper platform isderived asequation 10. with respect to time asvc =Jte - Jt F-1 Fe9 全 Jv918.oc = vc =b1 +b2 +b3314.Jre - Jr F-1Fewhere Jv R6X 6 is called the velocity Jacobian ma- trix which relates the joint space with the Cartesian space. Subscript v is attached to J to emphasize theBy chain rule, the following relationship is obtainedfrom 14. Sovelocity relationship.cc。 o。 ovc =9 + 全 J。 9。 te 9 + Jt 15.3.1. SingularitiesThe singular configuration is one of the most signifi- cant and critical problems in the design and control of parallel mechanisms.21 Unlike a serial mecha-where9 = e1e2e3d1d2d3 Tand = 12nism, operation of a parallel mechanism near a3 T. The subscript t of J represents translational motion. The rotational angular velocity = x ysingular point can cause severe destruction of the mechanism. Singularities of the Eclipse can be clas-z Tof the upper platform is derived using thesified into two types: differential singularity androtation matrix R = n s aand the skew sym-actuator singularity.13,2 If F is not of full rank, themetricity of RRT in ref. 14, p. 163mechanism is said to be at a differential singularity, while if Jv fails to be full rank, then the mechanism is said to be at an actuator singularity. At actuator。 n z。 sz。 azn y 。 9 + sy 。 9 + a y 。 9x。 n。 s。 asingularities, the end-effector cannot resist an ap-plied force, and the forward kinematics has multi-= n x x+ s+ a x9ple solutions. The actuator singularities occur at 30yz 。 9z 。 9z 。 9and 600 of the tilting angle, respectively. To elimi-。 n。 s。 az yyynate these singularities, two revolute joints are ad-3n x 。 9+ sz 。 9 + ax 。 9ditionally actuated.。 n z。 sz。 az3.2. Force Transmissionn y 。 + sy 。 + a y 。 In this subsection, using the principle of virtual。 n x+ nz 。 + s。 n y。 sxz 。 + a。 sy。 axz 。 。 a ywork, the relationship between the actuating forces of six sliding joints and the force-moment of theupper platform is derived. The six actuating forcesare denoted by f = f1 f2f3 f4f5 f6 T. The forcen x 。 + sz 。 + ax 。 and moment of the upper platform are denoted byF = FxFy Fz Tand M = MxMy Mz T, respec-全 Jre e+ Jr 16.tively. Let T 全 FTMT T. Let the generalized coordi-nates representing the position and orientation ofthe upper platform be u 全 Xc YcZc exey ez T.the substitution of 23. and 24. into 18. and 22.,Now, by using the incremental notation,comes8 u = Jv 8 918.be-19.respectively, yieldsvc= JvW1 .925.where 8 u = 8 Xc8 Yc8 Zc8ex8ey8ez Tand 8 9F.= 8e1, 8e 2 , 8e 3 , 8 d1, 8 d2 , 8 d3 T. Also, the applica- tion of the principle of virtual work to the mecha-M = Jf W2f26nism yields:Also, the split o position andf T 8 9 = TT 8 uThe substitution of 19. into 20. results in TT Jv - f T . 8 9 = 020.21.orientation parts yieldsvc =Jvo J o927.Because the virtual displacements of the generalized coordinates in 21. are independent, the following relation should holdFJFo f=MJMo 28.where Jv 0 , J o , JF o , JM o R3X 6and subscript ovT = J T .-1 f全 Jf f22.represents output analysis. Also, since all analysis procedures for four weighted Jacobian matrices in 27. and 28. are the same, we carry out only one analysis as a representing one. Let p and q denote awhere Jfis the force Jacobian matrix which mapsnormalized-input and an output vector, respec-the actuating forces in the joint space to the force-moment of the upper platform in the Cartesianspace. Jv and Jf are now used for the analysis of the input-output transmissivity and the manipulabilityof the Eclipse because they characterize the outputtively. That is, 27. and 28. are represented by a single equation. Then,q = Jo pcharacteristics of the upper platform with respect towhere J J , J, J , J4, q v , , F, M4, and pa given input velocity and force.ov o o F o M oc 9, f4.Since the six columns of Joare not linearly4. INPUT -OUTPUT TRANSMISSIVITYindependent, p for a given q is not unique. Instead, the minimum norm solution with respect to a given4.1. Output Analysisq is utilized. Let J+be the pseudoinverse of J.23oTo analyze the output characteristics of a mecha-ooThen,nism, the unit-norm inputs are often used. How- ever, these unit-norm inputs may not permit thep+= J+q29.actual operating range of the mechanism because the maximum velocities and forces of actuators may differ. Therefore, individual actuator velocities and forces are normalized using the weighting matrices as22The pseudoinverse, J+ , can be obtained from theosingular value decomposition asoo ooJ+ = H I+U T30.9 全 W-1 923.where Uo = u1u 2 u 3 is the eigenvector matrix ofooo ooo1Jo JT R3X3and Ho = h1 h 2h 3 h 4 h5h6 is the2f 全 W-1 f24.eigenvector matrix of JT J R6X 6. U and H are orthogonal matrices. Also, I+ R6X 3 is the trans-where W1 = diage1 max , e2 max , e3 max , d1 max , d2 max ,pose of the diagonal matrix Io R3X 6 which isd3 max .and W2 = diag f1 max , f2 max , f3 max , f4 max ,composed of the reciprocal singular values of Jo .f5max , f6max . denotes normalized values. Hence,Because the actuator input vector phas beennormalized, the Euclidean norm of p should satisfy the following inequality:obtained by substituting the above relationship into34. as -1 j2 222 /1p 1 2 = pT p 豆 131.u2 1d =1u2 2d+2u2 3d+336.Also, p+ should satisfy 31. because it is the mini-mum norm solution. Therefore, the substitution of29. into 31. yields:owhere u d 全 U T u d .oop+ .T p+= qT J+T J+ .q 豆 1 Hence, the substitution of 30. into 32. yields32.4.2. Input AnalysisIn contrast to the previous subsection, the ranges of input that achieve the following output ranges are analyzed. Then,qT J+T J+ .q = qT H I+U T .T H I+U T .q1v 1 豆 V1 1 豆 37.ooo ooo oocc maxmax= U T q. I+T I+ . U T q. 豆 1Toooo33.1F 1 豆 Fma x1M 1 豆 Mmax38.o1231Now, let q 全 U T q, where q = q qq T. Let ,Similar to the previous section, normalized outputs are defined as2 , and 31 2 3 . be the singular values ofJo . Thus, the following ellipsoid equation is derived from 33. So,1vc = R-1 vc2 = R-139.qq2212 2 + 2q23+ 2 豆 13434.F = R-1 FM = R-1 M40.123Equation 34. is now named the output manipula-where R1 = diagVc max , Vc max , Vc max ., R 2 = di- ag max , max , max ., R3 = diag Fmax , Fmax , Fmax ., and R4 = diag Mmax , Mmax , Mmax . Therefore, thebility ellipsoid for the weighted Jacobian matrix Jo .substitution o18.and22.,The above development is summarized as fol- lows: For all possible normalized inputs satisfying1p 1 豆 1, the maximum output is achieved in therespectively, and the split of Jacobian matrices intofour parts yieldsdirection of the first singular vector u1 of Uo which corresponds to the maximum singular value 1. Therefore, the maximum magnitude of output is 1and to achieve this maximum output, the input pshould be applied in the direction of h1 = J+ 1u1 .vc=FJvi J iJFi 941.oSimilarly, the minimum output is achieved in thedirection of the third singular vector u 3 of Uo which=fMJMi 42.corresponds to the minimum singular value 3. Thewhere J , J , J , and J R3X 6. The subscript imagnitude of this output is 3and the appliedvi iF iM irepresents input analysis. As a representative of 41.oinput p in this case is h 3 = J+ 3 u 3 . Therefore, theand 42., the following single equation is used.output range for all inputs satisfyingcomes:1p 1 豆 1 be-Then,q = Ji p43.3 豆 1q 1 豆 1 .35.Because the output vector q is normalized, the Eu-Finally, let u d be the desired output direction in the Cartesian space. Let be the maximum input-output transmissivity in that direction defined byclidean norm of q satisfies the following inequality:odq 全 U T u . Then, the maximum transmissivity is1q 1 2 = qT q 豆 144.Similarly, Jiis decomposed into three parts usingrotational manipulabilities are not distinguishable.the singular value decomposition asiJi = Ui Ii H T45.where Ii is a diagonal matrix composed of singularTherefore, 49. may not be suitable for the Eclipse.First, the condition number and the volume ofthe output manipulability ellipsoid for a givenweighted Jacobian matrix are defined as9values. The substitution of 43. and 45. into 44.yields:CN 全ma x: condition numbermi nmr m j2m50.TTiii Ji p. Ji p. = H T p. IT Ii . H T p. 豆 146.MEV 全r 1 + 2 /口 i :i=1Let p 全 H T p and , , and . bei123123the singular values of Ji. Thus, the following ellip- soid equation is obtained from 46. So,manipulability ellipsoid volume51.p21-2p22+ -2p23+ -2 豆 147.where m is the dimension of the manipulabilityellipsoid and r . is the Gamma function. The con-dition number represents the directional characteris-tics of the weighted Jacobian matrix. The larger the123Equation 47. is called the input manipulability el- lipsoid for weighted Jacobian matrix Ji. Therefore, the input range for all possible outputs satisfying1q 1 豆 1 becomes:13-1 豆 1p 1 豆 -148.condition number is, the severer the directional characteristics are. The larger the volume of the manipulability ellipsoid is, the greater the total out- put is for a given input. Therefore, it is desirable to have a small condition number and a large volume. The following new manipulability measure is intro- duced. Then,MEV J .5. A NEW MANIPULABILITY MEASUREm 全 CN J .: manipulability measure local. 52.Intuitively, a manipulability can be defined as how easily and uniformly the end-effector is able to move in arbitrary directions. To analyze a manipu- lability of the mechanism, the manipulability ellip-Now, the following individual manipulability mea- sures are definedMEV Jv .soid is the most intuitive and useful measure. The manipulability ellipsoid can be made by mapping am1 全:CN Jv .unit sphere in the joint input space to the output space through the Jacobian matrix. The major andtranslational velocity manipulability TVM.MEV J .minor axes of the ellipsoid indicate the directions inwhich the tool can move most and least easily andthe ease is proportional to the principal axes length.m2 全: CN J .Also, the magnitude and direction of the major and minor axes can be obtained from the singular valuerotational velocity manipulability RVM.MEV J .decomposition as described in Section 4. If the ellip-soid is larger and more circular, then the upperplatform has faster and more uniform motion. Fromthese physical insights, Yoshikawa5 defines the ma-m3 全FCN JF .MEV JM .: force manipulability FM.nipulability of a mechanism as1 2nM = det JJ T . = 49.m4 全: moment manipulability MM.CN JM .where subscripts v, , F, M represent the line velocity, angular velocity, force, and moment, re-In 49., however, the constraints on joint velocitiesand forces are not included and translational andspectively. Note that the above manipulability islocal, i.e., the value is different at each point in theworkspace. Therefore, if we integrate 52. over the entire workspace, the following global manipulabil- ity can be definedMi =HW mi r , l, R. dWHW dWi = 1, 2, 3, 453.where W denotes the entire workspace. Note that lowercase mi are used for local manipulabilities anduppercase Miare used for global manipulabilities.Finally, it is noted that these manipulabilities arefunctions of link parameters.If the velocity manipulabilities are large, thenthe mechanism can respond fast and furthermorethe response characteristics are uniform throughout the workspace. Similarly, if the force-moment ma-nipulabilities are large, the mechanism can resist large external disturbances and the resistance char- acteristics are isotropic.Figure 3 depicts the translational velocity ma- nipulability with respect to the radius of the upper platform and the length of a supporting link. Figure 4 shows two-dimensional cross-sections of Figure 3 for three different link lengths. The posture of the Eclipse for which the manipulability is calculated is such a position that the origin of the upper platformis oc = 0 0 1 and three rotational angles are = 00,户 = 900, and = 00. The radius of the lower platformis assumed to be 1. The maximum joint velocitiesand forces of the Eclipse are tabulated in Table III.As shown in Figure 4, the maximum manipula- bility is achieved when r ranges 04.-0.6. If a fastFigure 4. 2D cross-sections of Figure 3: l = 1.2, 1.6, and2.0.translational motion of the upper platform is de- sired, this translational manipulability measure be- come more important than others. Figure 5 shows a three-dimensional plot of the force manipulability measure at the same posture. Figure 6 shows two- dimensional cross-sections of Figure 5 for three dif- ferent link lengths. The force manipulability be- comes larger when the radius of the upper platform becomes larger and vice versa when the length of supporting link gets shorter.Figure 3. 3D plot of the translational velocity manipulability measure at oc = 00 1, = 00, 户 = 900, and = 00.Table III. Maximum joint velocities and torques of the Eclipse.JointsMaximumMaximumvelocitytorquerpm.N m.Three prismatic joints on200044the circular guideThree prismatic joints on450014.7the vertical columnTwo revolute joints on300023.3the supporting link6. OPTIMAL LINK DESIGNIn this section, two link parameters of the Eclipse are optimally designed by using the newly pro- posed manipulability measure in Section 5. The overall goal of this problem is to find suitable link parameters which can machine a workpiece as fast as it can and to maximize the force transmission. Assuming that the radius of the lower platform, R, is constant 30 cm., the optimal design problem is defined as follows:Optimization Problem: Rapid machining and maximum force transmissionFigure 6. 2D cross-sections of Figure 5: l = 1.2, 1.6, and2.0.where V r, l. is the workspace volume and Vr is the volume of a specific workpiece. Equation 54. is a constrained multicriteria optimization problem.Solution Using a Genetic Algorithm: It is very13Maximize M r , l . , M r , l . Tdifficult to solve54.analytically. Therefore, ther , lSubject to 5 cm 豆 r 豆 25 cm54.15 cm 豆 l 豆 60 cmV r , l . - Vr 0stochastic genetic algorithm of ref. 24, which doesnot require the differentiation of the objective func-tion, is utilized. This algorithm is particularly suit-able for solving complex multicriteria optimizationproblems. To select optimal link parameters of theFigure 5. 3D plot of the force manipulability measure at oc = 00 1, = 00, 户 = 900, and = 00.Eclipse, the following design criteria and fitness function are defined. So,Upper platform radius r . : 5 cm 豆 r 豆 25 cm Length of a supporting link l . : 15 cm 豆 l 豆 60 cm0,V r , l . V1 max3 maxr55.ellipsoid volumes associated with the four splitted Jacobian matrices, a new manipulability measure was defined. Two link parameters, the radius of the upper platform and the length of a supporting links,where 1and 2are weighting factors and 1 + 2were optimized by maximizing the new manipula-= 1. Vris assumed to be the volume of the cylinderbility measure introduced. The analysis methodsof radius 10 cm and height 10 cm. M1maxandand results obtained in this article can be easilyM3max are the maximum values of the translationalvelocity manipulability and the force manipulabilityfunctions, respectively. These values can be ob-tained by optimizing fitness functions M1 and M3 , respectively. Figure 7 shows the mean fitness values achieved as the algorithm generates. The optimallink parameters finally obtained arer = 14.355 cml = 39.169 cmMaximum fitness: 0.89917. CONCLUSIONIn this article, the forward and inverse kinematics, velocity and angular velocity transmissivity, force and moment transmissivity, and manipulability of aextended to the design problem of general 6-DOFparallel mechanisms.This work was supported in part by the Brain Korea 21 Program of the Department of Education, Korea. The first author would like to thank Professor Jong Won Kim and Professor Frank C. Park in
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