Comparative Analysis of Characteristics of the Coupled and Decoupled Parallel Mechanisms

CHINESE JOURNAL OF MECHANICAL ENGINEERING●468●Vol. 23, No. 4, 2010DOI: 10.3901/CJME.2010.04 468, available online at www.cjmcnet.com; www.cjmenet.om.cnComparative Analysis of Characteristics of the Coupledand Decoupled Parallel MechanismsZENG Daxing *, HOU Yulei, LU Wenjuan, and HUANG ZhenCollege ofMechanical Engineering, Yanshan Universiy, Qinhuangdao 066004, ChinaReceived November 21, 2009; revised Jume 17, 2010; accepted June 23, 2010; publisbed lelroicallyy August 20, 2010Abstract: The existence of coupling makes the parallel mechanism possess some special advantages over the serial mechanism, while itis just the coupling that brings about the parallel mechanism some limitations, such as complex workspace, high nonlinear relationshipbetween input and output, difficulties in static and dynamic analysis, and the development of control system, which restricts isapplication fields. The decoupled parallel mechanism is currently one of the research focuses of the mechanism fields, while the studyon the different characteristics between the decoupled and coupled parallel mechanisms has not been reported. Therefore, this paperperforms the systematic comparative analysis of the 3-RPUR and the 3-CPR parallel mechanisms. The features of the two mechanismsare described and their movement formns are analyzed with screw tbeory, The inverse and forward displacement solutions are solved andthe Jacobian matrices are obtained. According to the Jacobian matrices and by using the theory of physical model of the solution space,the workspace, dexterity, velocity, payload capability, and siffness of the mechanisms are analyzed with pltting the indices atlases. Theresearch results prove that the effects of the coupling on the parallel mechanism are double-side, and then the adoption of the decoupledparallel mechanism should be determined by the requirements of the concrete application situation. The contents of this paper should beuseful for the type synthesis and practical application of the parallel mechanism.Key words: prallel mechanism, couple, decouple, screw theory, index atlasThen the decoupled parallel mechanism (DPM), in which1 Introductionthere exists a one-to-one correspondence relationshipbetween the input and output variables, attracts manyJust as mentioned on philosophy, things are always the scholars' interetsh-s. In this paper, the kinematic pairs willunity of opposites, and this is also confirmed in the be denoted with following symbols: P for prismatic pair, Rresearch process of mechanism. When the serial for revolute pair, U for universal pair, and C for cylindricalmechanism is forging ahead in its development, the parallel pair. HUANG and LI4 invented a fully-isotropic 3-CPRmechanism (PM), in which the fixed base and the moving three-degrees-of-freedom (3-DOF) decoupled translationalplatform are connected by two or more limbs, becomes the PM. CARRICATO, et al, discussed the general problem ofresearch focus of the intemational academic and the topological synthesis and classification of translationalengineering field gradually.PMs, and investigated both their constraint and directCompared with the serial mechanism, the parallel singularitiesls, and presented a pointing parallel mechanismmechanism has some outstanding properties in certain U-PUR PRRU with fully decoupled degrees of freedom0.aspects, such as a large payload to weight ratio, KONG and GOSSELIN proposed a 3-CRR 3-DOFconsiderable sifess，, low inertia, high dynamic translational parallel manipulator with linear actuators']performnances, and so on. The advantages of the PM are and three classes of input-output decoupled parallelclosely related to the coupling between the limbs. While it manipulators with 2 to 4 DOFs using a geometricis no other than coupling that makes the PM possess some approach8. KIM and TSAI9I introduced a 3-DOFdisadvantages, including smaller workspace, lower Cartesian parallel manipulator and performed its optimaldexterity, more complex control command, and multiple design. L, et alo], presented a 3-DOF translational parallelsingularities. The difficulties in the structural design, static manipuR rIIRF wurhirh 1decoupled motionand dynamic analysis and development of control systemof in t中国煤化工ted sigulritifreethe PM are also caused by the existence of the coupling, fully-ifYHCN M H GPOF and proposed awhich restrict the applications of the PM.methoa Ior suructural synunesis oased on the theory oflinear transformations.Generally speaking, the DPM not only keeps the电Corresponding author. E mail: roboms@ysu.cdu.cnThis pojet is uppore by Natinal Naturl Scince Foundation of principal advantages of the PM, but also avoids the mainChina (Grant No. 50875227)CHINESE JOURNAL OF MECHANICAL ENGINEERING●469.drawbacks of the general coupled PM. To reveal theThe coordinate system 0rxyz(i=1, 2, 3) is built with itsseparated characteristics of the coupled and decoupled PMs,origin located on the center of the R pair of the ith limbtaking the 3-RPUR PM and the 3-CPR PM as example connected with the fixed base, in which y axis is consistentrespectively, this paper focuses on their comparative with the axis of this R pait, and Z axis is perpendicular toanalysis.the fixed base plane. Selecting the limb ABiC1 as example,The organization of this paper is as follows: Following we can obtain the screw system of the limb in 01-xVZI asthe introduction, the structure composition of the 3~RPUR follows:PM and the 3-CPR PM are described and their modes ofmotion are analyzed in section 2. Displacement analysis,&=(01 0;00 0),including the forward and inverse displacement solution, is$|2=(000;a 0 b),performed in section 3. Section 4 concerns performance$z=(010;c0d),.(1)analysis, including workspace, dexterity, velocity, payload. S4=(t0l;e f g),capability and siffness etc. The paper is concluded in$s=(t0l;hpq),section 5, summarizing the present work.2 Mechanism Features Analysiswhere a, b,c,d, t,l, e,f, g, h, p and q are all finite non-zeroreal number.As shown in Fig. 1, the 3-RPUR PM is composed of anThe reciprocal screw of the screw system in Eq. (1) isupper platform, a lower platform and three uniform limbsconnecting the two platforms with R, P, U and R in turn.$=(000;l 0 -1).(2)connection points are located on a circumcirclesymmetrcally 120° apart. The axis of the R pair connectedIt can be seen that $ expresses a constraint couplewith the lower platform is parallel with one of the two axeswhich is perpendicular to the plane formed by the twoof the U pair, the motion direction of the P pair isrotational axes ofU pair of the limb AB:C.perpendicular to these two parallel axes, and the other axisSimilarly, the reciprocal screws S2 and S of theof the U pair is parllel with the axis of the R pair other two limbs A.B;C2 and As.BsC3 can be solved, and theyconnected with the upper plaform. Taking the upper andare also perpendicular to the plane formed by the twolower platfrm as the moving platform and the fixed base rotational axes of∪pair of the crresponding limbrespectively, we set up the moving coordinate system 0~xyzrespectively. Obviously, $，S路，and S arenon-and the stationary coordinate system O-XYZ with theircoplanar which accounts for that they are lineaorigins located at the geometrical center of the upper andindependent. Therefore the three rotations of the movinglower platform rspetiely. In adition, the P pairs of three platform are constrained compltely, that is, the mechanismlimbs are selected as the input pairs. The circumcircles ofonly has three translations.the upper and lower platform are denotedas r and R ,The mobility can also be obtained by the modifiedrespectively. The distance between A:(i=1, 2, 3) and B;Kutzbach-Gribler criterion!12. What should be noted is that(i=l, 2, 3) is denoted as 4(i=1, 2, 3), and the length of the the 3-RPUR PM has not common constraint, passive DOF,pole B;C(i=I, 2, 3) is supposedas L.and redundant constraints, and then its DOF isisicM =d(n-g-1)+'之sf+y-s=G6(11-12-1)+15+0-0=3,(3)B,where M is the DOF of the PM; d the order of themechanism, and d=6-h; n the common constraint;i3n denotes the number of links; g the number ofkinematic pairs; fr the feedom of the ith pair; v thenumber of redundant constraints; 5 the passive degree of小freedom.Fig. 2 shows the schematic diagram of the 3~CPR PMwith th中国煤化工-:onnects the movingplatfion: pair and the C pair,respectMYHC N M H Gparallel with that ofthe C pair, and the motion direction of the P pair, whichlocates between the R pair and the C pair, is perpendicularFig. 1. Schematic diagram of 3-RPUR 3~DOF PMto these two parallel axes. In addition, the axes of the C●470.ZENG Daxing, et al: Comparative Analysis of Characteristics of the Coupled and Decoupled Parallel Mechanismspairs of the three limbs keep perpendicular one another. are three redundant constraints, so this PM only has threeThe stationary coordinate system 0-XYZ is set up with its translations.origin located at the intersection of the three C pairs, andBy the modified Kutzbach-Gribler criterion, the DOF ofthe X, Y and Z axis are collinear with the axis of the threeC the 3-CPR PM can be solved aspairs respectively. The moving coordinate system 0-xyz isset up with its origins located at the geometrical center ofthe triangle B\B2B3, and the x, y and z axes are parallel withM=d(n-g-1)+2 Cf+v-s=that of the stationary coordinate system respectively. The6(8-9-1)+12+3-0=3.(6)translation of the three C pairs are selected as the input, andthe input quantity are denoted as LI， L21 and L13 Displacement Analysisrespectively. The vertical distance from the origin of thecoordinate system 0-xyz to the axis of the R pair connecting3.1 Inverse displacement solutionwith the moving platform is denoted as m, and the length ofFor the 3-RPUR PM, the coordinates of points A and Cthe link A,B;(i=1, 2, 3) is denoted as L;(i=1,2, 3).(i=1,2,3) with respect to 0-XYZ can be expressed asfollows:A(R 0 ),A(-R/2√3R/2 0),A(-R/2 -√3R/2 ),l3C(q+r Yo zo),|BC2(x-r/2 xo+√3r/2 zo), .BSx0-C(x-r/2 yo-√3r/2 2o). .21On one hand, the traectory of point B;(i= 1, 2, 3) locatesLL3Ion a circle of which takes C;(i=1, 2, 3) as the centre and Las radius, on the other hand, B; also locates on a circle ofwhich takes A(i=1, 2, 3) as the centre and l(i=1, 2, 3) asxradius. Tben the input parameter 么，h and Lj can beFig. 2. Schematic diagram of3-CPR 3-DOF PMobtained asTaking the limb AjB\ as example, we can obtain its screwh=+220(x +r- R)+system as follows:&|=(00 0;1 0 ),&n2=(100;00 0),-+7)(4)&;3=(0 0 0;0 a b),&4=(1 00;0 c d),1:=[i+(- R2+(-5y_-到的+(5-where a, b, c and d are all finite non-zero real number.Then the reciprocal screws of the screw system of the√3R+%+zr-R-号+-y-z0|4+z2，(8limb A.B: can be solved asS=(00 0;01 0)，$i=(0 00;001). (5)=i +(-R}P +(+53y-号+(5R-Itcan be seen that S{ and $2 express a constaint5部)》+4(-R-号号-2)r+9)couple around axis Y and axis Z respectively, whichconstraints the rotation of the moving platform around the wheretwo axes.Similarly, the reciprocal screws $2 and S2 of the中国煤化工r -2(J3元-20.limb A2B2 which around X axis and Z axis respectively, andMYHCNMHG$}1 and 852 of the limb AzB3 which around X axis and Yaxis respectively, can also be solved.When derivatives are taken with respect to time for Eqs.Then it is obvious that these six reciprocal screws (7), (8) and (9), the relationship between the input andconstrain all the rotation of the moving platform and there output velocity can be expressed with matrix form asCHINESE JOURNAL OF MECHANICAL ENGINEERING●471●(iu iz is)(划)站|=|j21 jn 323|)，(10)(zo) (js ix2 i)(zo)(13)001)where3.2 Forward displacement solutionx+r-R+2-y%2For the 3-RPUR PM, the forward displacement solution2needs to solve the simultancous Eqs. (7), (8) and (9) withj=known (, 2, 1). Nevertheless, it is comparative dificultto obtain the closed solution by using curent methods.6至-为-r+R,艺-y-2yoWhile for the 3-CPR PM， if the driving inputVL-y2(L1, L21, L31) is determined, the location coordinates of21the moving platform are_2z0-12(2 -y2)而=41-m，yo=L21-m， Zo=L31.(14)242n=(R-√3xo-r-x)-(30 +V3p)xFrom the above derivation, it can be seen that, for the3-RPUR PM, the input-output equations are highly coupledand the expression of Jacobian matrix is complex, which(r-R-号22will result in the difficulty in its inverse and forwarddisplacement solution, and bring out some disadvantages加=(3(-x- R)+xo+=2A-(3x+o)xfor the performance analysis. While the Jacobian matrix ofthe 3-CPR PM is a constant matrix, and the inverse andforward displacement solution of this mechanism is sor-R-号simple to solve, which is convenient for further researchjz3=20-A12and practical application.4 Perfromance Analysisian=(R+3yo-r-=o)--(3x-3x)x4.1 Normalization of structural parameters(-R-血_↓The theory of physical model of the solution spacel1-15provides a tool for the research of the relationship betweenV3Bthe performance evaluation indices and the structuralia =√3(x+R-n)+Yo-+(↓3x-y)xparameters. By using the performance indices atlases, wecan determine the domain of structural parameters withgood values of the indices conveniently.(-气- 211/2>The three limbs of the 3-RPUR PM possess the same2z0-B12structure and distribute symmetrically, then without2/consideration of the twist angle between the axis of the pairand the moving platform or the fixed base, this mechanismA=√82-2(3x+)o),has three variable parameters r，R and L. For theB=. V82-2(V3x0-)0)*.simplification consideration, we normalize the structuralparameters with linear dimensions as follows:If matrix J is not singular, the Jacobian matrix of the. r'=r/R, R'=R/R, l'= L/R,(15)3-RPUR PM can be solved asJ=术(11)where R =(r+R+ L)/3.Eq. (15) should subject toFor the 3-CPR PM, the relationship between the inputr'+R'+L'=3. 0≤r'.R'.L'≤3.(16)and output can be directly obtained from Fig. 2, that is中国煤化工In thvalues of structural41=x+m，L21=yo+m, L31=zo.(12) paramelMYHCNMH Gposedas ,=0.6,R'=L'=1.2.Taking derivatives with respect to time for Eq. (12), theFor the similar consideration, the dimensionless inputJacobian matrix of the 3-CPR PM can be obtained asparameters of 3-RPUR PM can be expressed as●472●ZENG Daxing, et al: Comparative Analysis of Characteristics of the Coupled and Decoupled Parallel Mechanismsdescribed as shown in Fig. 4. This workspace is just like a1=h/R，|21 =h/R，41=;/R.(17) cube of which each side is 1, and there does not existsingular point in the whole workspace.For the 3-CPR PM, there is only one variable parameters By comparing Fig. 3 and Fig. 4, it can be seen that them, which does not infuence the output and the Jacobian available workspace of the 3-CPR PM is bigger than that ofmatrix of the PM without consideration of the twist angles the 3-RPUR PM, so the application fields of the formerbetween the axes of pairs.should be wider than the ltter fom this point.For the comparability between the 3-RPUR and the3-CPR PM, the same variable range of the input is set asfollows:吊3.62.5≤你b2n, 631, L, L2, L1≤3.5.(18)4.Workspace analysisAs one of the important kinematics indices, the size ofthe workspace determines the range of motion of thes 2.6mechanism. For a PM with good application value inpractical engineering, the shape of its workspace should beregular, and the mechanism should have better kinematicsperformances and do not exist singular configuration in the2426283.03.2whole workspace.NondimensiorAccording to Eqs. (7), (8), (9), (15) and (16), amushroom workspace of the 3-RPUR PM is obtained andFig. 4. Workspace of the 3-CPR PMshown in Fig. 3. The workspace distributes symmetrically120° apart and the thickness of the center is about 0.6.4.3 Dexterity analysisGenerally speaking， the condition number of theJacobian matrix is taken as the performance evaluationindex for the dexterity analyisllo, and the bigger the1.2 tcondition number is, the worse the dexterity is. The.0-condition number of the Jacobian matrix can be definedasll].6_k, _ (J),(19)σmin(J)'where σmx(J) and σmim(J) are the maximum andminimum singular values of Jacobian matrix, respectively.According to Eqs. (10), (11), and (19), the dexterity2021.5 -1.0 -0.atlases of the 3-RPUR PM are shown in Fig. 5.Noimisioallengh郑Nondimensional height 2o=3.6Nondimensional height 2o=3.7(a) Upper surface1.Sr1..5-/0.5-05-0.5-3.21.5-1.0-0.50 0.51.01.s-1.0-0.50 0.5 1.01.5Nondumensional length xNondimensional length x号3.Nondimensional height z0= =3.9Nondimensional height zg=4.01.0 - 0570.5? 1.5r? L.5p0 05) 0.1.0-中国煤化工((b) Lower surfacefYHCNMHGFig. 3. Workspace of the 3-RPUR PM-1.0-0.50 0.51.01.5z_.σ0.500.51.oOn the basis of the theory of physical model of theNondimensional length xoNondimensional length和solution space, the workspace of the 3-CPR PM isFig. 5. Dexterity atlases of 3-PRUR PM.CHINESE JOURNAL OF MECHANICAL ENGINEERING●473●In fact, the condition number defined in Eq. (19) onlydescribes the dexterity of the mechanism in a certain pose,Sr欧x」。dWGrmin =f。dW， (23)to give an overall evaluation on the dexterity of a certainworkspace, the global conditioning index, proposed byGOSSELIN, et all8), is defined as follows:where 6vmax and Svmin are the maximum and minimumglobal velocity indices, respectively; W denotes thef⊥dwWworkspace of the mechanism.Jwkjη=(20)i dWNondimensional height 2z=3.6_ Nondimensional height z0 = 3.7* 1.5厂1.5复1.0-复1.0where刀j denotes the global conditioning index, and0.50.5-/0<ηj≤1; W denotes the workspace of the mechanism.oKThe global conditioning index is esentally the average-0.-0.5 Fof the reciprocal of the condition number of the Jacobian-1.00matrix in a certain workspace. The bigger ηj is, the是-1.0-0.500.51.01.5 z -1.0-0.50 0.51.01.5higher the dexterity of the mechanism is.Nondimensional length xoNondimensional length xAccording to Eq. (20), the global conditioning index ofNondimensional height z0= 3.9 Nondimensional height z0= 4.0the 3-RPUR PM can be calculated out, which is equal to0.046 414 and is far smaller than 1. This shows thathe吾1o1.1.sdexterity of this mechanism is very poor.5057While for the 3-CPR PM, according to Eq. (13), thecondition number of the Jacobian matrix is equivalent to 1，which indicates that the global conditioning index is equalto 1 and keeps invariable in the whole workspace. So it can享-1.0 -0.50 0.51.01s z -1.0 -0.50 0.51.01.sbe said that the 3~CPR PM possesses isotropic kinematicNondtmensional length x0Nondimensional length xqperformances and its dexterity is very good.(旧) Minimum velocity index alases4.4 Velocity analysisNondimensional height zg= 3.6Nondimensional height Z0= 3.7Denoting the input and output velocity vector of the3-RPUR PM as θ and V respectively, we have唱l.0r0.5-0.5- .o-ofV= J6,(21)-0.5--1.0-where v=[xjozo]", 0=[[J',and J is4.0-050 0.51.0-1.0 : 0.500.510Jacobian matrix.Nondimensional length xη .When|日= 1, the velocity extremum, which is usuallytaken as the evaluation index for the output velocity of theNondimensional height 2z0=3.9Nondimensional height z0= 4.0mechanism, can be defined asl18]1.0r三0.5-|wuuI= r.(苏), am-=_-.历)，(22) .-0.5-1.0where| Vmax and Vmin are the maximum andminimurm of the output velocity vector, respectively;Nondimensional length与hma(J"J) and Iymio(J'J) are the maximum and(b) Maximum velocity index alasesminimum characteristic values of matrix JTJ , respectively.Fig. 6. Atlases of the velocity index of the 3-RPUR PMAccording to Eqs. (10), (11) and (22), the velocity indexatlases of the 3-RPUR PM are plotted and shown in Fig. 6.Then the maximum and minimum global velocity indicesBeing similar to the condition number discussed above, of the| 中国煤化工space can be solved,the velocity extremum only describes the maximum or whichminimum of the velocity of the mechanism on a certain OwjMHCNMHG3-CPRPMisaunitpoint. For the purpose of evaluating the velocity matrix, the maximum and minimum of the output velocityperformance in a certain workspace, the global velocity vector are both equal to 1, and keep invariable in theindicesll9l are defined as follows:workspace. Therefore the maximum and minimum global●474●ZENG Daxing. et al: Comparative Analysis of Charateristics of the Coupled and Decoupled Parallel Mechanismsvelocity indices of the 3-CPR PM are also equivalent to 1.is far bigger than that of the 3-RPUR PM, while theFrom the above analysis, it can be seen that the maximum global payload capability index of the 3-RPURminimum global velocity index of the 3-CPR PM is bigger PM is a lttle bigger than that of the 3-CPR PM. So thethan that of the 3-RPUR PM, while the condition of the coupled 3-RPUR PM can be used for the machine whichmaximum global velocity index is just opposite to that of requires large payload capability in some directions andthe minimum one. So the coupled 3-RPUR PM is more small payload capability in the other directions, and thesuitable for the application which requires high velocity in decoupled 3-CPR PM can be used for the situationsome directions and low velocity in the other directions, requiring the uniform payload capability in all directions,and the decoupled 3-CPR PM is more suitable for the and should possess wider application fields.situation requiring the uniform velocity in all directions.Nondimensional height z0= 3.6Nondimensional height Z0 = 3.74.5 Payload capability analysisThe relationship between the generalized force vector喜1.0-是1.0-f and the extermal force vector F applied to the end0.50.5 li -erofactuator of the mechanism can be expressed as-0.5--0.5F=Gf,-1.0-1.0-0.500.5 1.0 T.s-1.00.50 0.51.01.5where G is the force Jacobian matrix, and G=(JT)-'.Nondmensional length xNondimensional length xWhen |f|= l, the extremum of payload capability canNondimensional height 20= 3.9Nondimensional height 20= 4.0be defined as followsl20]:1.sp复1.0摹.1.0个多|Fal=√pm (G'G), |Fual=√xm(GG)，(25)0.5-/0.s/js -057-0.5 t.分whereI Fmx and|Fmin| are the maximum and-10F2minimum of the payload capability in a certain pose,-1.0-0.50 0.51.0 1.srespectively; Apmx(G'G) and hmin(G'G) are theNondimensional length xqmaximum and minimum characteristic values of matrix(a) Minimum velocity index atlasesG'G, respectively.Nondimensional height Z0= 3.6Nondimensional height 2=3.7According to Eqs. (10), (11) and (25), the payloadcapability index atlases of the 3-RPUR PM are plotted and1.0复1.0rshown in Fig. 7.0.The maximum or minimum obtained fom Eq. (25) onlyo-denotes the payload capability of the mechanism when themoving platform locates on a certain position in theworkspace, then to describe the payload capability in the. -1050.51.010500510whole workspace, the global payload capability indices areNondimensional length x0defined as followslol.Nondimensional height动= 3.9Nondimensional height zo= 4.0f. IFm |dwf. |mmwWSFmax =fi dW”，SFminfdW(26) .0.5-0叶0-where SFmax and SFmin are the maximum and minimum-1.0-Loglobal payload capability indices， respectively; WNondimensional length为denotes the workspace of the mechanism.Then the maximum and minimum global payload(b) Maximum velocity index atlasescapability indices of the 3-RPUR PM in the wholeworkspace can be solved, which are equal to 1.865 7 and中国煤化工pability idx0.091 338, respectively. The maximum and minimumglobal payload capability indices of the 3-CPR PM are both 4.6YHCNMH Gequivalent to 1.The relationship between the deformation D of the endFrom the above analysis, it can be seen that the actuator and the general force vector F applied to theminimum global payload capability index of tbe 3-CPR PM mechanism can be expressed asCHINESE JOURNAL OF MECHANICAL ENGINEERING.475.D=CF=JJF,(27)The maximum or minimum deformation obtained fromEq. (28) only reflects the deformation situations of thewhere C is the flexibility matrix, and C = JJT .mechanism in a certain position, then to evaluate thWhen |F|=VFTF=1 ，the extremum of thestiffness performance of the mechanism in the wholedeformation can be solved according to the followingworkspace, the global stiffness indices are defined asfollowslI4":expressions|pul=√mma(C'C), |\.l=0o(CTC), (28) .Gomaf. dw'where |IDmax | andI Dmin| are the maximum and(29)minimum deformation of the mechanism in a certain pose,f IDm[dwrespectively; Apmr(c"C) and Apmim(C'C) are thef dWmaximum and minimum characteristic values of matrixc'c , respectively.whereandDminare the maximum and minimumAccording to Eqs. (10), (11) and (28), the sifness index global sifness indices, respectively; W denotes theatlases of the 3-RPUR PM are poted and shown inFig. 8. workspace of the mechanism.Then the maximum and minimum global stiffnessNondimensional height z0= 3.6 Nondmensional height Z0= 3.7 indices of the 3-RPUR PM in the whole workspace can besolved, which are equal to 1 228.5 and 0.300 38,看1.0F20.24二respectively. The maximum and minimum global stiffness01//520.327三0.5-p -032.indices of the 3-CPR PM are both equivalent to 1.0From the above analysis, it can be seen that the-0.5maximum global stiffness index of the 3-RPUR PM is farbigger than that of the 3-CPR PM, while the minimum-1.0050 0..0T.s2 Hσ05δ 0510'sglobal sifness index of the 3-RPUR PM is smaller thanNondimensional length xqNondmensional length xthat of the 3-CPR PM. This indicates that the coupledNondimensonal height z0 = 3.9 Nondimensional heught Zq= 4.0?3-RPUR PM is more suitable for the application requiring1.0[ 09-028七1.0「安large siffness in some directions.0..5Fe5 Conclusions-0.5--1.0-. 唱(1) Taking the 3-RPUR PM and the 3-CPR PM as2 510-0530 osor's艺T0-0.50 0.51.01.sexample, the comparative analysis of characteristics of theNondumcnsional lengh xNondimensonal length xcoupled and decoupled PMs, including the kinematics,(a) Mnimum stfness index atlasesworkspace, dexterity, velocity, payload capability andNondimensional height 20= 3.6Nondmensional height 0=3.7 stifiness, has been performed.1.5p(2) Compared with the coupled PM, the Jacobian matrix看1.0of the DPM is a constant matrix, which is independent of0.5-0.5个the pose of the mechanism. The condition number of theJacobian matrix is equal to 1, and the mechanism possessesexcellent force and motion transmission capabilities.0一o-(3) Both the forward and the inverse displacement-1.0 0 1osolutions of the DPM are straightforward and almost do notNondimensional length xoNondmensional length对Nondmensional height 20=3.9Nondimensional height Eo = 4.0need to calculate. The shape of the workspace of the DPMis regular, and it is free from singularities in the whole1.0workspace..5F(4) The velocit, payload capability and sifness indices0F (of the coupled PM are better than that of the DPM in somedirectiindices of the DPM,includ中国煤化工dd cabpbil and-1.05 TosifneHCN M H Gace.Nondumensional lengh xoNondimensional length x(5) The higher the decoupling extent of the PM is, the(b) Maximum sifness index atlaseseasier the kinematics and dynamics analysis are, which isFig. 8. Atlases of the sifness index of 3-PRUR PMmore favorable for the control and the improvement of theZENG Daxing, et al: Comparative Analysis of Characteristics of the Coupled and Decoupled Parallel Mecbanismsmovement precision.Mechanism Conference, St Louis, MO, USA, 1983: 26 33.(6) The coupled and decoupled PMs bave their own[15] YANG Jihou. The space model and dimensional types of thefour-bar mechanisms[J]. Mechanism and Machine Theony, 1987,advantages and disadvantages, and which one is more22(1): 71-76.suitable for a certain application should depend on the[16] POND G, CARRETERO J A. Fomulaing Jacobian matrices forspecific situation.the dexterity analysis of parallel manipulators(J]. Mechanism andMachine Theory, 2006, 41(12): 1 505-1 519.References刀] SALISBURY J K, CRAIG JJ Articulated hands: force control and[1] PATARINSKIS P, UCHYAMA M. Psitn/orientation decoupledkinematic issues(J]. 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ZENG Daxing, bom in 1978, is curently a lecturer in Yanshan7] KONG Xianwen, GOSSELIN C M. Kinematics and singularity University, China. He received his PhD degree on mechatronicanalysis of a novel type of 3-CRR 3-DOF translational parallel engineering in Yanshan University, China, in 2008. His researchmanipulator[J] The International Journal of Robotics Research, interests include parallel mechanism, type synthesis and image2002, 21(9): 791- -798.processing.8] KONG Xiaowen, GOSSELIN C M. Type synthesis of iputoutput Tel: +6-335-8059817; E-mail: roboms@ysu edu.cndecoupled parallel manipulators([]. Transactions of the CanadianSociery, for Mechanical Engineering 2004, 28(2): 185- -196.HOU Yulei, bom in 1980, is currently a lecturer in Yanshan9] KIM H s, TSAI L W. Design optimization of a Cartesian paralelUniversity, China. He received his PhD degree on mechatronicemanipulator[C)/Proceedings of the ASME Design Engineeringengineering in Yanshan University, China, in 2007. His researchTechnical Conference, Montreal, Qucbec, Canada, September 29,interests include force sensor, parallel mechanism and optimal2002, 5B: 865 872.[10] LI Weimin, GAO Feng, ZHANG Jianjun. R-CUBE, a decoupleddesign. .Tel: +86-335-8059817; E-mail: ylhou@ysu.edu.cnparallel manipulator only with revolute joints[J]. Mechanism andMachine Theony, 2005, 40(4): 467-473.[1] GOGU G. Flly-isotropic over-constrained parallel wrists with two LU Wenjuan, borm in 1983, is currently an asant in Yanshandegrees of freedom[C]/Proceedings of the 2005 IEEE International University, China. She received her master degree on mechanicalConference on Robotics and Automation, Barcelona, Spain, 2005:design and theory in Yanshan University, China, in 2008.[12] HUANG Zhen, ZHAO Yongsheng, ZHAO Tishi. Advanced spatial4 025- 4 030.E-mail: wenjuan_ Ju@163.commechanism[M]. Bejing: Higher Education Press, 2006. (in Chinese)HUANG Zhen, borm in 1936, is currently a professor and a PhD[13] DAVIES T H, BAKER J E, THOMPSON A G R. A finite,candidate supervisor in College of Mechanical Engineering,3- dimensional alas of 4-bar linkages([]. Mechanism and MachineYanshan University, China. His main research interests includeTheory, 1979, 146); 389- 403.[14] BARKERC R. WEEKS G A. A physical model of the solutionparallel robot, type synthesis and topology.space for fowr bar mechanisms[CY/Proceedings-OSU AppliedTel: +86-335-8074709; E-mail: huangz@ysu.edu.cn中国煤化工MYHCNMHG