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A key point dimensional design method of a 6-DOF parallel manipulator for a given workspace Rui Cao 1, Feng Gao,1, Yong Zhang1, Dalei Pan1 State Key Lab of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, PR China a r t i c l ei n f oa b s t r a c t Article history: Received 3 April 2014 Received in revised form 7 November 2014 Accepted 8 November 2014 Available online 25 November 2014 This paper presents a new method of dimensional design for a 6-PSS parallel mechanism according to a given workspace. A symmetrical description has been found to describe the 6-D workspace concisely for the dimensional design. Many key point characteristics have been found and verifi ed by the kinematic analysis and the method of Lagrange multipliers. Furthermore,thedirectrelationsbetweenthegivenworkspaceandthemanipulatorsgeometrical parameters have been derived. The proposed design method which is based on these key point characteristics has very high effi ciency and accuracy. Additionally, the avoiding of the complex analysis of the manipulators workspace and the dimensionless derivation make the possibility of wide use of this method. 2014 Elsevier Ltd. All rights reserved. Keywords: Parallel manipulator Dimensional design Workspace 6-PSS Key point 1. Introduction The interest for parallel manipulators arises from the fact that they have better load-carrying capacity, better stiffness, and better precision than serial manipulators 14. Thus the research on designing parallel manipulators has become a hot topic in the international robotic research area 59. The design of parallel manipulators is a challenging problem in the machinery product design process. The type synthesis is for designing the confi guration for manipulators 1012. And then the geometrical parameters ofmanipulatorsshouldbedetermined bythedimensionaldesign.Becausethetypesof parallelmechanisms arealmostunlimited,the dimensionaldesignmustbebasedonacertaintypeofmechanisms.Theparameterdesignmethodspresentedinreference13,14are based 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, the dimensional 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. The fi rst one uses many points to describe the given workspaceand then check whether the manipulator with certain parameters fi ts the design requirements at each point 1517. The other one establishes a function between the parameters and the workspace boundaries of the manipulator, then make sure that the given workspace is within the manipulators workspace boundaries 1822.Basedonseveralkeypointsthatwehavefoundinthisstudy,thispaperattemptstoexploreanewwayofdimensionaldesign for 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 confi guration has been proposed. Compared with the traditional 6-SPS parallel manipulators, this 3-3-PSS parallel manipulator allows higher isotropy of the manipulators performance, larger rotation range of the moving platform and less body inertia. Mechanism and Machine Theory 85 (2015) 113 Corresponding author. E-mail addresses: azuresilent (R. Cao), gaofengsjtu (F. Gao), crlycf (Y. Zhang), pandalei100new (D. Pan). 1 P.O. Box ME290, Mechanical Building, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, PR China. /10.1016/j.mechmachtheory.2014.11.004 0094-114X/ 2014 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: Tobeginthedesign,therequiredworkspaceshouldbeclearlydescribed.Becausethe6-dimensionalworkspacecannotberepresent- edgraphicallyinahuman-readablewayandtherearenogeneralwaytoanalyticallydeterminetheboundariesofthe6-Dworkspacefor 6-DOFparallelmanipulators,mostliteratures2327dividethe6-Dworkspaceintopositionworkspaceandorientationworkspace.The position workspace refers to a space that the manipulators moving platform can reach with a certain orientation. And it can be easily depicted.Theorientationworkspaceisthecollectionofalltheorientationsthatthemovingplatformcanachieveatacertainpoint.How- ever, due to the complexity of the rotating angles, the orientation workspace is diffi cult to be determined and represented. Considering the 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 3 shows 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 analysis Thearchitectureofthenew3-3-PSSparallelmanipulatorisshowninFig.1,whichiscomposedofamovingplatform,afi xedbase, and six supportinglimbswith identical geometrical structure. The limbs are numbered from 1 to 6. Each limbconnects the fi xed base tothemovingplatformbyaprismaticjoint,asphericaljointBiandasphericaljointAiinseries.Alinearactuatoractuatestheprismatic joint of each limb along a fi xed 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 symmetry axis of the manipulator are the same, and here we use the parameter a to represent this distance. The other three linear actuators of the limbs 4, 5, and 6 are arranged with their axes vertically. The centers of the joints A1 A6of the moving platform are distributed symmetrically on a circle of radius a. The center of this manipulator is at the intersection of the plane PBand the symmetry axis of the manipulator, on which attached a fi xed Cartesian reference coordinate frame Ox, y, z. The fi xed frames y-axis and z-axis are in the plane PB, and its x-axis coincides with the symmetry axis of the manipulator. A moving frame O x , y , z is attached on the moving platform at point O which is the center of the circle that points A1 A6located on. Considering the fact that the manipulator is axisymmetric, let point O coincides with point O when the moving platform is at the initial position. Thus the workspace of the manipulator is also axisymmetric with respect to the fi xed frame O. Before designing the geometrical parameters of the manipulator, the required workspace should be clearly described. As can be seen from the previous discussion, concisely describing the required 6-D workspace is a challenging problem. In this research, for the orientation description of the moving platform, only the pointing vector (showed in Fig. 2) rather than the rotation about its symmetry axis is concerned. In fact this has the same situation for many machine tools. Based on this, we use a special set of Euler angles to represent the orientation of the moving platform. The moving platform fi rst rotates about the fi xed x-axis by an angle-, thenaboutthefi xedz-axisbyanangle,andfi nallyaboutthefi xedx-axisbytheangle(Fig.2).Andwecansimplywritetherotation matrix for this case as: R Rot x;Rot z;Rot x; ccsss css2 c2cs scc sscs ccsc2 s2c 2 4 3 5; 1 Fig. 1. The confi guration of the proposed 3-3-PSS parallel manipulator. 2R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113 where c stands for cosine, s stands for sine, , and 0, , respectively. For the convenience of description, Eq. (1) can be abbreviated as R r11r12r13 r21r22r23 r31r32r33 2 4 3 5: 2 And it can be observed that r32 r23 sin 2 1cos 2 :3 This special set of Euler angles gives anintuitive representation of themovingplatforms orientation.The pointingvector is decid- ed byand . Duetothesymmetryof themanipulator, itis easytofi ndoutthat therange ofis unlimitedwhile isnot.Thus,all the possiblepointingvectorsthatthemovingplatformcanachieveatacertainpointconstituteacone.Andtheapertureoftheconeisonly related 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, we describethegivenworkspaceasacylinderwithradiusofRc,andheightof2Hc.Additionally,themanipulatorshouldhavethepointing dexterityof m at any pointwithin this cylinder. This human-readable workspacedescription fi tsforthe manipulators symmetry and makes the design objective clearly. Knowing that this workspace description is actually 5-DOF, to represent a 6-DOF workspace, an additionaldexterityindexoftherotationaboutthemovingplatformssymmetryaxis isneeded.In thissituation,themovingplatform should fi rst perform an additional rotation about the fi xed 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 clarifi ed. The following part will fi nd this out by analyzing the kinematics of the manipulator. As the six limbs of the manipulator have identical geometrical structure, we can choose one typical limb for the analysis and its vectors are described in Fig. 3. The linear actuators axis is represented by eiwhich is a unit vector. The direction of the rigid link is represented by liwhose magnitude is Li. The vector between O and the center of the joint Aiis represented by ai with respect to the moving frame O , and aiwith respect to the fi xed frame O. It can be found from the previous part that the magnitude of ai/aiis a. When the manipulator at the initial position thatmentionedabove,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 fi xed frame can be described by the position vector of the moving platform p = PxPyPzT, and the rotation matrix R. Thus the generalized coordinates 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:4 In some cases, the joints Biand Aiwhose stiffness are the lowest of the manipulator need a strong structure to increase their stiffness. However, the strong structure always limits the rotation ranges of these joints. Therefore, the swing amplitude of the Fig. 2. The pointing dexterity and the special set of Euler angles. 3R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113 rigid link should be studied. We defi ne the angle between liand eias the joint angle Biof joint Bi. As joint Ai is fi xed on the moving platform,thedefi nitionshouldwithrespecttothemovingframeOx,y,z.ThusthejointangleAiofjointAi isdefi nedastheangle between liand Rei. Biand Aiare depicted in Fig. 3. The following equations about Biand Ai can be achieved by their defi ni- tion: li? ei LicosBi5 li? Rei LicosAi:6 According to these defi nitions, the rotation of the rigid link about its own axis liis not involved. So Biand Airepresent the swing amplitudeoftherigidlinkwithrespecttotheconnectingjoint.ThemaximumvaluesofBiandAiareveryimportantforthedesignof the spherical joints and meaningful for avoiding the interference between the rigid links. The six limbs canbedividedintotwogroupsaccordingtotheconfi guration ofthemanipulator.Thelimbs1,2,and3are contained in group 1, and the limbs 4, 5, and 6 in group 2. These two groups have different kinematic characteristics, thus need to be studied separately. 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, a fi xed 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?T and ci 0Lia?T. Assume that lilxlylz ?T . Substituting all the known variables into Eq. (4) yields the following equations: lx ar13 px7 ly ar23 pyLi qi8 lz ar33 a pz:9 Furthermore, the following relation can be achieved with the fact that Liis the magnitude of vector li: ly ? ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi L2 il 2 xl 2 z q :10 According to Eq. (10), lyhas two possible solutions. When a coordinate of the moving platform makes Li 2 lx 2 lz 2 b 0, lyhas no solution, which means that this coordinate is out of the manipulators reachable workspace. The situation Li 2 lx 2 lz 2 = 0 means that the moving platform reaches the boundary of the reachable workspace. This situation is singular and should be avoided in Fig. 3. One typical limb of the manipulator. 4R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113 practice. Because of these, the sign of lyshould be constant during the operation of the manipulator. Let p = 0 and R = I when the moving platform at the initial position. Substituting them into Eq. (8) yields ly a ? 0 0Li0 Lib0:11 Therefore, 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 1 qi ar23py Li ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffi L2 ar13 px2 ar33 a pz2 q 12 lican be written with Eq. (7), Eq. (9), and Eq. (10) as li ar13 px ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi L2 il 2 xl 2 z q ar33 a pz 2 6 4 3 7 5: 13 Then we can obtain li? ei ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi L2 i ar13 px 2 ar33 a pz2 q ;14 li? Rei my mx mz:15 Where, 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 fi xed Cartesian reference coordinate frame Oaix, 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,a 0 i 0 0a?Tandci Li0a?T with respect to the frame Oai. Substituting all the known variables into Eq. (4) yields the following equations. lx ar13 pxLi qi16 ly ar23 py17 lz ar33 a pz18 and the following relation can also be obtained with the fact that Liis the magnitude of vector li lx ? ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi L2 il 2 yl 2 z q :19 Similartotheanalysis 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 Li ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi L2 i ar23 py ?2 ar33 a pz2 r :20 With Eq. (16), Eq. (17), and Eq. (18), lican be written as li ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi L2 il 2 yl 2 z q ar23 py ar33 a pz 2 6 4 3 7 5: 21 Then we can obtain li? ei ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi L2 i ar23 py ?2 ar33 a pz2 r ;22 li? Rei m2x m2y m2z ? :23 5R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113 Here, 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 independent and 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 p