三自由度并聯(lián)機(jī)器人的結(jié)構(gòu)設(shè)計(jì)與運(yùn)動(dòng)學(xué)分析【說(shuō)明書(shū)+CAD+SOLIDWORKS】,說(shuō)明書(shū)+CAD+SOLIDWORKS,三自由度并聯(lián)機(jī)器人的結(jié)構(gòu)設(shè)計(jì)與運(yùn)動(dòng)學(xué)分析【說(shuō)明書(shū)+CAD+SOLIDWORKS】,自由度,并聯(lián),機(jī)器人,結(jié)構(gòu)設(shè)計(jì),運(yùn)動(dòng)學(xué),分析,說(shuō)明書(shū),CAD,SOLIDWORKS A key point dimensional design method of a 6-DOF parallelmanipulator for a given workspaceRui Cao1, Feng Gao,1, Yong Zhang1, Dalei Pan1State Key Lab of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, PR Chinaa r t i c l ei n f oa b s t r a c tArticle history:Received 3 April 2014Received in revised form 7 November 2014Accepted 8 November 2014Available online 25 November 2014This paper presents a new method of dimensional design for a 6-PSS parallel mechanismaccording to a given workspace. A symmetrical description has been found to describe the 6-Dworkspace concisely for the dimensional design. Many key point characteristics have beenfound and verified by the kinematic analysis and the method of Lagrange multipliers.Furthermore,thedirectrelationsbetweenthegivenworkspaceandthemanipulatorsgeometricalparameters have been derived. The proposed design method which is based on these key pointcharacteristics has very high efficiency and accuracy. Additionally, the avoiding of the complexanalysis of the manipulators workspace and the dimensionless derivation make the possibilityof wide use of this method. 2014 Elsevier Ltd. All rights reserved.Keywords:Parallel manipulatorDimensional designWorkspace6-PSSKey point1. IntroductionThe interest for parallel manipulators arises from the fact that they have better load-carrying capacity, better stiffness, and betterprecision than serial manipulators 14. Thus the research on designing parallel manipulators has become a hot topic in theinternational robotic research area 59. The design of parallel manipulators is a challenging problem in the machinery productdesign process. The type synthesis is for designing the configuration for manipulators 1012. And then the geometrical parametersofmanipulatorsshouldbedetermined bythedimensionaldesign.Becausethetypesof parallelmechanisms arealmostunlimited,thedimensionaldesignmustbebasedonacertaintypeofmechanisms.Theparameterdesignmethodspresentedinreference13,14arebased on 6-DOF Gough-type manipulators and 3-DOF parallel manipulators, respectively.Generally, one of the most important design objectives is to let the manipulator work in a given workspace. Therefore, thedimensional design of parallel manipulators for a given workspace is an important problem, which has not gained too much interest.So far, there are mainly two ways to design the geometrical parameters of parallel manipulators according to a given workspace. Thefirst one uses many points to describe the given workspaceand then check whether the manipulator with certain parameters fits thedesign requirements at each point 1517. The other one establishes a function between the parameters and the workspaceboundaries of the manipulator, then make sure that the given workspace is within the manipulators workspace boundaries1822.Basedonseveralkeypointsthatwehavefoundinthisstudy,thispaperattemptstoexploreanewwayofdimensionaldesignfor a new 6-DOF parallel manipulator according to a given workspace. This design method is fast and its result is accurate.In our previous work, a new type of 6-DOF parallel mechanism with an orthogonal 3-3-PSS configuration has been proposed.Compared with the traditional 6-SPS parallel manipulators, this 3-3-PSS parallel manipulator allows higher isotropy of themanipulators performance, larger rotation range of the moving platform and less body inertia.Mechanism and Machine Theory 85 (2015) 113 Corresponding author.E-mail addresses: (R. Cao), (F. Gao), (Y. Zhang), (D. Pan).1P.O. Box ME290, Mechanical Building, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, PR China.http:/dx.doi.org/10.1016/j.mechmachtheory.2014.11.0040094-114X/ 2014 Elsevier Ltd. All rights reserved.Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: workspace refers to a space that the manipulators moving platform can reach with a certain orientation. And it can be easilydepicted.Theorientationworkspaceisthecollectionofalltheorientationsthatthemovingplatformcanachieveatacertainpoint.How-ever, due to the complexity of the rotating angles, the orientation workspace is difficult to be determined and represented. Consideringthe symmetry of our parallel manipulator, a concise way of describing the 6-D workspace is found for the dimensional design.The paper is organized as follows. Section 2 introduces the modeling of the design problem and the kinematics analysis. Section 3shows how the key point characteristics are found. The design method and its application are discussed in Section 4. Finally,concluding remarks are presented in Section 5.2. Modeling of the design problem and kinematic analysisThearchitectureofthenew3-3-PSSparallelmanipulatorisshowninFig.1,whichiscomposedofamovingplatform,afixedbase,and six supportinglimbswith identical geometrical structure. The limbs are numbered from 1 to 6. Each limbconnects the fixed basetothemovingplatformbyaprismaticjoint,asphericaljointBiandasphericaljointAiinseries.Alinearactuatoractuatestheprismaticjoint of each limb along a fixed rail. Between the joint Biand joint Aiis a rigid link of length Li(i=1,6).The three linear actuators of the limbs 1, 2, and 3 are arranged with their axes located in a horizontal plane PB, and the angles be-tween each of their axes are 120 while these axes do not intersect at one point. The distances between these axes and the symmetryaxis of the manipulator are the same, and here we use the parameter a to represent this distance. The other three linear actuators ofthe limbs 4, 5, and 6 are arranged with their axes vertically. The centers of the joints A1 A6of the moving platform are distributedsymmetrically on a circle of radius a. The center of this manipulator is at the intersection of the plane PBand the symmetry axis ofthe manipulator, on which attached a fixed Cartesian reference coordinate frame Ox, y, z. The fixed frames y-axis and z-axis are inthe plane PB, and its x-axis coincides with the symmetry axis of the manipulator. A moving frame O x , y , z is attached on themoving platform at point O which is the center of the circle that points A1 A6located on. Considering the fact that the manipulatoris axisymmetric, let point O coincides with point O when the moving platform is at the initial position. Thus the workspace of themanipulator is also axisymmetric with respect to the fixed frame O.Before designing the geometrical parameters of the manipulator, the required workspace should be clearly described. As can beseen from the previous discussion, concisely describing the required 6-D workspace is a challenging problem. In this research, forthe orientation description of the moving platform, only the pointing vector (showed in Fig. 2) rather than the rotation about itssymmetry axis is concerned. In fact this has the same situation for many machine tools. Based on this, we use a special set of Eulerangles to represent the orientation of the moving platform. The moving platform first rotates about the fixed x-axis by an angle-,thenaboutthefixedz-axisbyanangle,andfinallyaboutthefixedx-axisbytheangle(Fig.2).Andwecansimplywritetherotationmatrix for this case as:R Rot x;Rot z;Rot x;ccssscss2 c2cs sccsscs ccsc2 s2c2435;1Fig. 1. The configuration of the proposed 3-3-PSS parallel manipulator.2R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113where c stands for cosine, s stands for sine, , and 0, , respectively. For the convenience of description, Eq. (1) can beabbreviated asR r11r12r13r21r22r23r31r32r332435:2And it can be observed thatr32 r23sin 2 1cos 2:3This special set of Euler angles gives anintuitive representation of themovingplatforms orientation.The pointingvector is decid-ed byand . Duetothesymmetryof themanipulator, itis easytofindoutthat therange ofis unlimitedwhile isnot.Thus,all thepossiblepointingvectorsthatthemovingplatformcanachieveatacertainpointconstituteacone.Andtheapertureoftheconeisonlyrelated to the maximum range of which is represented by m. We call mas the pointing dexterity index of the moving platform.To take advantage of the symmetry of the manipulator, we restrict the required workspace as a symmetric space. Hence, wedescribethegivenworkspaceasacylinderwithradiusofRc,andheightof2Hc.Additionally,themanipulatorshouldhavethepointingdexterityof mat any pointwithin this cylinder. This human-readable workspacedescription fitsforthe manipulators symmetry andmakes the design objective clearly. Knowing that this workspace description is actually 5-DOF, to represent a 6-DOF workspace, anadditionaldexterityindexoftherotationaboutthemovingplatformssymmetryaxis isneeded.In thissituation,themovingplatformshould first perform an additional rotation about the fixed x-axis by an angle , and the rotation matrix can be written as Rot(x,)Rot(z,)Rot(x, -)Rot(x,). However, 5-DOF is enough for our current study and most multi-DOF machine tools.After the analysis of the required workspace, what parameters of the manipulator need to be determined should be clarified. Thefollowing part will find this out by analyzing the kinematics of the manipulator. As the six limbs of the manipulator have identicalgeometrical structure, we can choose one typical limb for the analysis and its vectors are described in Fig. 3. The linear actuatorsaxis is represented by eiwhich is a unit vector. The direction of the rigid link is represented by liwhose magnitude is Li. The vectorbetween O and the center of the joint Aiis represented by ai with respect to the moving frame O , and aiwith respect to thefixed frame O. It can be found from the previous part that the magnitude of ai/aiis a. When the manipulator at the initial positionthatmentionedabove,ei(i= 1,2,3)isperpendiculartoai,itshouldbenoted.AndtheinitialpositionofBiinthissituationisrepresent-ed by point Ciwhose position vector is ci. With the special set of Euler angles, the transformation from the moving frame to the fixedframe can be described by the position vector of the moving platform p = PxPyPzT, and the rotation matrix R. Thus the generalizedcoordinates of the moving platform can be described as (Px, Py, Pz, , , 0).Let qirepresent the stroke of the linear actuator. Then we can simply get the following relation from Fig. 3:li p Ra0iqieici:4In some cases, the joints Biand Aiwhose stiffness are the lowest of the manipulator need a strong structure to increase theirstiffness. However, the strong structure always limits the rotation ranges of these joints. Therefore, the swing amplitude of theFig. 2. The pointing dexterity and the special set of Euler angles.3R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113rigid link should be studied. We define the angle between liand eias the joint angle Biof joint Bi. As joint Aiis fixed on the movingplatform,thedefinitionshouldwithrespecttothemovingframeOx,y,z.ThusthejointangleAiofjointAiisdefinedastheanglebetween liand Rei. Biand Aiare depicted in Fig. 3. The following equations about Biand Aican be achieved by their defini-tion:li? ei LicosBi5li? Rei LicosAi:6According to these definitions, the rotation of the rigid link about its own axis liis not involved. So Biand Airepresent the swingamplitudeoftherigidlinkwithrespecttotheconnectingjoint.ThemaximumvaluesofBiandAiareveryimportantforthedesignofthe spherical joints and meaningful for avoiding the interference between the rigid links.The six limbs canbedividedintotwogroupsaccordingtotheconfiguration ofthemanipulator.Thelimbs1,2,and3are containedin group 1, and the limbs 4, 5, and 6 in group 2. These two groups have different kinematic characteristics, thus need to be studiedseparately. For the sake of symmetry, the rigid links in one group should have the same length. In group 1 for i = 1, 2 and 3, afixed Cartesian reference coordinate frame Oaix, y, z is attached at the point O. For simplicity and without losing the generality,we let its y-axis point in the negative direction of the vector eiand let its x-axis coincide with the x-axis of the frame Ox, y, z.With respect to the frame Oai, it can be known from the architecture of the manipulator that ei 010?T, a0i 00a?Tand ci 0Lia?T. Assume that lilxlylz?T. Substituting all the known variables into Eq. (4) yields the followingequations:lx ar13 px7ly ar23 pyLi qi8lz ar33 a pz:9Furthermore, the following relation can be achieved with the fact that Liis the magnitude of vector li:ly ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zq:10According to Eq. (10), lyhas two possible solutions. When a coordinate of the moving platform makes Li2 lx2 lz2b 0, lyhas nosolution, which means that this coordinate is out of the manipulators reachable workspace. The situation Li2 lx2 lz2= 0 meansthat the moving platform reaches the boundary of the reachable workspace. This situation is singular and should be avoided inFig. 3. One typical limb of the manipulator.4R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113practice. Because of these, the sign of lyshould be constant during the operation of the manipulator. Let p = 0 and R = I when themoving platform at the initial position. Substituting them into Eq. (8) yieldsly a ? 0 0Li0 Lib0:11Therefore, Eq. (10) should take a negative sign. Then substitute Eq. (10) into the left side of Eq. (8) and we can get the inverse so-lution of the actuating stroke qiof group 1qi ar23py LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 ar13 px2 ar33 a pz2q12lican be written with Eq. (7), Eq. (9), and Eq. (10) asliar13 pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zqar33 a pz264375:13Then we can obtainli? eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar13 px2 ar33 a pz2q;14li? Rei my mx mz:15Where, my= r22li ei, mx= ar12r13+ r12pxand mz= ar32r33+ r32a + r32pz.In each limb of group 2 (i = 4, 5, 6), for simplicity and without losing the generality, a fixed Cartesian reference coordinate frameOaix, y, z is also attached at the point O with its z-axis intersecting eiand its x-axis coinciding with the x-axis of the frame Ox, y, z.Hence, it can be observed from the architecture of the manipulator thatei 100?T,a0i 00a?Tandci Li0a?Twith respect to the frame Oai. Substituting all the known variables into Eq. (4) yields the following equations.lx ar13 pxLi qi16ly ar23 py17lz ar33 a pz18and the following relation can also be obtained with the fact that Liis the magnitude of vector lilx ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2yl2zq:19Similartotheanalysis of group 1, we can obtain that Eq. (19) should take a negative sign.Substitute it into the leftside of Eq. (16),and we can get the inverse solution of qifor group 2.qi ar13px LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar23 py?2 ar33 a pz2r:20With Eq. (16), Eq. (17), and Eq. (18), lican be written asliffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2yl2zqar23 pyar33 a pz264375:21Then we can obtainli? eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar23 py?2 ar33 a pz2r;22li? Rei m2x m2y m2z?:235R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113Here, m2x= r11li ei, m2y= ar21r23+ r21pyand m2z= ar31r33+ r31a + r31pz.Through the analysis above, it can be found that among the manipulators geometrical parameters, only a and Liare independentand need to be determined. And the maximum ranges of qi, Bi, and Aineed to be found out for the manufacture of the manipulator.Thus the design problem can be summarized as follows.Requirements:1. The manipulator should achieve the given workspace which is a cylinder with radius of Rc, and height of 2Hc. And the manip-ulator should have the pointing dexterity of mat any point of the given workspace. In some cases, when the requiredworkspace is complicated, the given workspace can be described as many coaxial cylinders with different pointing dexterityindices as Fig. 4 shows.2. Insomesituations,themaximumrangesofBiandAiarelimitedforthepurposeofincreasingthestiffnessofthejointsoravoidingthe interference.Design task:1. Find out appropriate geometrical parameters of a and Lithat can let the manipulator meet with all the requirements listed above.2. Afterthedeterminationofthegeometricalparametersabove,findoutthemaximumrangesofqi,Bi,andAiforthemanufactureofthe actuators and joints.3. The key point characteristics of the manipulator within the given workspaceIn this research, though a large amount of calculations, we have found that the maximum ranges of qi, Bi, and Aialwaysoccur at some certain locations in spite of the dimension changing of the given workspace nor the manipulator. In otherwords, there are some certain relations between these locations and the given workspace. This characteristic is very helpfulfor the dimensional design, thus we call these locations as key points. This section will prove the existence of these key pointstheoretically using the method of Lagrange multipliers and establish the relations between the key points and the givenworkspace.Forthesakeofgeneralization,wemaketherelatedparametersofthedimensionaldesigndimensionlessbylettingeachofthembedivided by Rc. Thus the workspace cylinders dimensionless radius is 1, and its dimensionless height is 2H. Where H = Hc/Rc. As a re-sult, the results of the dimensional design based on this dimensionless workspace cant be treated as the geometrical parameters ofthe manipulator directly, unless multiply each of them by Rc(it should be noted that the angles are not affected in this procedure).The two limb groups have different key point characteristics due to the configuration of the manipulator. So the characteristics ofthe two limb groups should be studied separately.Fig. 4. A description of the given workspace.6R. Cao et al. / Mechanism and Machine Theory 85 (2015) 1133.1. Group 1 (i = 1,2,3)As the given workspace is a cylinder, the points in the given workspace must meet the following equations:p2y p2z1;24HpxH:25From theanalysis in Section 2, we know that lyinEq. (10)should have a solution to let themanipulator reach the current positionand orientation. So it can be derived thatL2iN ar13 px2 ar33 a pz2:26The maximum value of therightpart of Eq. (26) represented by is studied. Andthe parameters involved are px, pz, and . Theirconstraint equations can be written asg1 p2y p2z1g2 pxHg3 pxHg4 mg5 8:27Thus the Lagrange function can be written as 1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:28The extreme values of occur where the gradient of K is zero. The partial derivatives arepx 0;py 0; 0; 0k 0;k 1;5k 2kk 0;k 1;58:29By solving the equation system (Eq. (29) and comparing the extreme values, we can obtain that reaches its maximum valuewhen px H;pz 1; 2; m, or px H;pz 1; 2; m. Substituting these two solutions into Eq. (26) yieldsL2iN asm H2 acm a 12:30Toensurethatthemanipulatorcanreacheverypointofthegivenworkspace,aandLishouldbechosentolettheEq.(30)establish.Andthegeneralizedcoordinateofthekeypointsforthiscaseare H;1;2;m;0?and H;1;2;m;0?.Where meansthis valueis arbitrary.Inordertofindthemaximumrangeofqi,weneedtofindboththeminimumandmaximumvaluesofqi.AndtheLagrangefunctionfor finding its minimum value can be written as qi1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:31No