[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】球形机器人的并行稳健性设计-外文文献

orientasmechanismibleoption,freedom by replacing the passive revolute joints with self-aligning pin joints, which is equivalent to replacing the passiveMechanism and Machine Theory 46 (2011) 335343Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: /locate/mechmtrevolutes with spherical joints. The drawback of this approach is that unnecessary mobility is introduced, that invariablycompromises the stiffness of the robot. A means to cope with manufacturing errors by means of calibration is not an option inthe case at hand, because spherical linkages are overconstrained. This means that small machining errors leading to revoluteswith axes that do not intersect at one common point render the mechanism a hyperstatic structure, if one is ever capable ofassembling it.1In our design we replace the unactuated revolute joints by cylindrical joints, thereby introducing only those extra degrees offreedomwhicharenecessaryandsufcient.Hence,a displacementanalysisofthemechanismis needed todetermine notonlytheprecision machining is, of course, anusually included, to allow for the inevitablerotations of the revolute joints but also theassembly errors that render the revolute axesrather spatial, capable of producing almost Corresponding author. Tel.: +1 514 3984488; fax:E-mail addresses: .jo (K. Al-Widyan),1Assembling such a closed kinematic chain with manufacturing0094-114X/$ see front matter 2010 Elsevier Ltd.doi:10.1016/j.mechmachtheory.2010.11.002if one is willing to pay for it. As an alternative, extra degrees of freedom aremachining errors. The designers of the Agile Eye provided the extra degrees ofSpherical parallel robots are used totool beds and workpieces 1,2, as welldevelopment of the Agile Eye 3,4.Ideally, all the joints of a sphericaldue to machining errors, it is not poss 2010 Elsevier Ltd. All rights reserved.a rigid body in three-dimensional space. Applications include orienting machine-orienting a camera tracking fast-moving objects. The latter application led to theare revolutes, with their axes intersecting at a common point. However,to manufacture such a mechanism with conventional machining operations; high-Dual numbersPrinciple of TransferenceInverse-kinematics analysis1. Introductiondimensioned.The robust design of parallel spherical robotsKhalid Al-Widyana, Xiao Qing Mab, Jorge AngelesbaDepartment of Mechatronics Engineering, The Hashemite University, P.O. Box 150459, Zarqa 13115, JordanbDepartment of Mechanical Engineering & Centre for Intelligent Machines, McGill University, 817 Sherbrooke St. W. Montreal, Canada H3A 2K6article info abstractArticle history:Received 7 October 2009Received in revised form 10 November 2010Accepted 11 November 2010Available online 4 December 2010In this paper a robust methodology, reported in a previous work, is applied to the design of theunactuated joints of a spherical three-degree-of-freedom parallel robot, the Agile Wrist (AW).Robustness is needed because of the extremely difficult task of manufacturing a sphericalmechanism with all its joint axes concurrent at a single point. In order to account for theunavoidable manufacturing errors, it is proposed here to replace the unactuated revolute (R)joints of an existing design by cylindrical (C) joints. The latter function even in the presence ofnonconcurrent axes. A procedure, based on dual numbers, is used to solve the inversekinematics of the entire mechanism in order to determine the rotations and translations ofeach C joint. Based on statistical results of the kinematic analysis, the C joints are suitablyKeywords:Agile WristSpherical parallel robotsStatistical analysistranslations at the unactuated cylindrical joints, in the presence of machining andnon-concurrent. Obviously, the resulting robot will not be spherical anymore, butspherical displacements.+1 514 398 7348.xqmacim.mcgill.ca (X.Q. Ma), angelescim.mcguill.ca (J. Angeles).errors is possible if the links are deformed when assembling the chain.All rights reserved.In this paper, a stochastic approach is followed to represent inaccuracies in the dimensions of the robot links at hand. Then, aprocedure is developed to estimate the above-mentioned cylindrical-joint displacements via an inverse-kinematics analysis.The analysisof the linkage under study boils downto solving a system of trigonometric equations. The establishedapproach tosolving this system relies on the trigonometric tan-half-angle identities transforming the equations at hand into polynomialequations. However, this transformation suffers from singularities at solutions close to , and hence, the said approach is notrobust. To cope with this problem, a robust geometric approach tonding the roots of the inputoutput (I/O) equation of four-barlinkages of the RCCC type was developed by Bai and Angeles 5. This approach was derived from that used for the analysis ofplanar and spherical RRRR linkages in the same reference.Replacingrevolute joints by their cylindrical counterparts results in changingthe topologyof therobot at hand,i.e., going froma spherical to a spatial linkage. The most straightforward way to derive the inputoutput equation for a spatial RCCC four-barlinkage is to apply the Principle of Transference 68. The beginnings of this principle are traced back to the early 1960s, one of itsmost notable applications having been reported by Yang and Freudenstein 9, who analyzed a RCCC spatial four-bar mechanismby dualizing the closure equations of a spherical four-bar mechanism.2. Kinematic analysis of the Agile WristThe Agile Wrist (AW),depictedin Fig.1a,is a three-degree-of-freedom robotdesignedastheterminalmoduleofan 11-degree-of-freedom long-reach robot 10. The AW is a parallel robot with three equal legs, a typical leg having the architecture of the336 K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343spherical serial robot shown in Fig. 1b. As shown in Fig. 1a, each leg couples the triangular plate, usually referred to as the movingplate (MP), executing a robotic task, with the plate on which the motors are mounted, commonly termed the base plate (BP). Infact, the AW was derived from the Agile Eye, as disclosed by Gosselin and his team 3,4. It is noteworthy that the proximal links ofthe Agile Wrist were designed as curved links with noncircular central curves, as reported in 11. The central curves weredesigned, in turn, following an optimization procedure under which the planar curve at hand was synthesized so as to exhibitminimum curvature values. Moreover, the same curve was designed so as to blend smoothly with the straight segmentsaccommodating the bearings and shafts of its end joints, of the R and C types. The Agile Wrist is used as a parallel robot to rmlyhold and orient a tool for tasks such as shot-peening.In the subsection below, we discuss the procedure developed to evaluate the cylindrical-joint translations via an inverse-kinematics analysis. The joint relative translations are important design parameters, for they allow the designer to allocate theminimum space needed to accommodate the joint translations arising by virtue of machining and assembly errors, under pre-scribed tolerances.2.1. The dual inverse kinematics of a RCCC chainA displacement analysis of each leg is conducted here, yielding the translations of the cylindrical joints.The problem reduces to the inverse kinematics of one leg of the Agile Wrist, which is a serial kinematic chain. We assume that,due to symmetry, each of the two other legs will produce the same results. A serial spherical wrist is shown in Fig. 1b; the chain athand is the same, except for the second and third revolute joints, which are now replaced with cylindrical joints, to accommodatethe offsets between neighboring axes. Hence, dual numbers can be used for this case, in order to not only determine the rotationsbut also the translations of the joints.The analysis at hand consists in nding the passive-joint displacements upon considering a xed pose of the moving platform,and hence, of the end-effector of Fig. 1b and working backwards to solve for the rotation of the actuated joint. In the ensuinganalysis, we resort to Fig. 1b, but regarding it as a RCCC chain instead. Now, the Principle of Transference states that the geometricFig. 1. The Agile Wrist: (a) its layout; and (b) one of its identical legs.relations of a spatial linkage can be derived by dualizing the counterpart relations for a spherical linkage.2Hence, by dualizing allvector and scalar variables of Fig. 1b, except for 1, we obtain the displacement analysis of the RCCC chain at hand. For a givenorientation of the end-effector, let the dual unit vector e3, in base-frame coordinates, be given aswhere12 iseffectordisplacemfromainto itsQiC138imatricewheretovectorcenterThewhichin Eq.whichOnand hence,with c337K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 3353432Dualizing is sometimes referred to as “putting hats” on all the variables and parameters in the spherical-linkage equations.vxs1s1vys1c1+ vzc1c2=0 8()cos() and s()sin(). The derivation of this equation and others needed in this context is described in detail in 13.e3C1381= QC1381QT33e3C1383|zo3where o3denotes the third row of Q3C1383.Ifqijdenotes the (i,j) entry of QC1381, then,e3C1381= q123+ q133q223+ q233q323+ q333C138T vxvyvzC2C3T7Upon substitution of Eqs. (4) and (7) into Eq. (3), one scalar equation is obtained for 1:Q1C1381Q2C1382= QC1381QT3hi36hiQ1C1381Q2C1382Q3C1383= QC1381whencee3C1381= Q1C1381Q2C1382e3C13835the other hand,1 1e2C1381= Q1C1381e1C1381= 1sin11cos11C138T4is nothing but the third column of QC138. Moreover,isderivedbyinspectionofFig.1b.Hence,acoordinatetransformationisneededinordertoexpressthetwovectorsinvolved(3) in the same coordinate frame, namely, F1, as explained below.eT2e3= cos23displacementanalysisofthesphericalrobotofFig.1bisoutlinedbelow.Tothisend,theDenavitHartenberg(DH)notationfollowed, with coordinate frame Fi, for i=1,2,3,4,xed to the (i1)st link, where 0 denotes the base and 3 the end-(EE) of the robot depicted in the foregoing gure. Moreover, a vector v represented in Fiis denoted vC138i. In the inverse-ent problem (IDP), the attitude of the EE is given by a rotation matrix R in Fi, the joint angles i13that will carry the EEreferenceattitudetoRbeingsought.Moreover,let QiC138idenotetherotationmatrixthat carriesvectorcomponents in Fi +1counterparts in Fi. We have, moreover, within this notation, eiC138ie =0;0;1C138T. For quick reference, the structure of thes is recalled:Qi QiC138icosiisiniisinisiniicosiicosi0 ii24352icosiandisini, with 1and 2shownin Fig. 1b, while 3is the angle made by the axis of the third revolute, parallele3,withZ4,theZ-axisofF4,attachedtotheEE.Thisaxisisuser-dened,itssoleconditionbeingthatitpassesthroughtheof the wrist. It is not shown to avoid overloading the gure.key step in solving the IDP of the spherical wrist at hand is setting up the relationspace, a dual unit vector represents a line in the same space.Thee3C1381= vxvyvzC2C3T1the subscript 1 refers to the base frame. We recall here that, while a real unit vector represents a direction in EuclideanUpon dualizing the foregoing expressions, we obtain the counterpart equation for the RCCC chain, namely,wherejoint beingThetwo solutions.Further,sphericalwhereDenavitcarryingdual partorthogonal33 identityMatrias 13where15 is338 K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 3353433The cross product matrix (CPM) D of the three-dimensional vector d is dened, for any vector v of the same dimension, as D = CPM d d v=v.c3= w1c2+ w2s215as3= w1c2s2+ w2c2c2+ w3s215brij= rij+ C15ij14ijbeing the (i,j) entry of matrix DR.From Eqs. (11a) & (11b), it is possible to solve for the translation d2of the second joint. Each of the foregoing equations is thendecomposed into two equations, one for the primal, one for the dual part of each dual equation. A similar approach is used toobtain the translation d3of the third cylindrical joint, which is attached to the moving platform; this is done by dualizing thecorresponding equations derived in 13 for spherical robots, namely,above equations ares 2s2= s3r12+ c 3r13c1+ s 3r22+ c 3r23s111as 2c2= c 1s 3r12+ c 3r13s1c 1s 3r22+ c 3r23c1s 1s 3r32+ c 3r33 11bi= i+ C15di, with didenoting the signed distance between consecutive normals of the joint axes, according with theHartenberg notation 13, while rij, for i,j=1,2,3, are the entries of the dual rotation matrixR= R + C15DR 12the end-effector from its reference pose to its current pose. The primal partRofRis a proper orthogonal matrix, while itsinvolves matrix D, which is the cross-product matrix3of the translation vector d. Moreover, matrixR is a dual propermatrixdet(R)=+1.The reference attitudeisdenedbyR =133andD = O33,where133andO33representtheand zero matrices, respectively.x R is specied by an angle of rotation and an axis of rotation, given by the unit vector e.ThismatrixisthenfoundR = eeT+ cos 133eeTC16C17+ sinE 13E is the cross-product matrix of e, thus obtaining the primal part of matrixR. Hence, each entry of the dual rotation matrixa dual number, namely,where rij, for i,j=1,2,3, are the entries of the rotation matrix R dening the orientation of the end-effector; the dual counterpartsof thes2s2= s3r12+ c3r13c1+ s3r22+ c3r23s110as2c2= c1s3r12+ c3r13s1c1s3r22+ c3r23c1s1s3r32+ c3r33 10binthepresenceofarootcloseto.Asanalternative,arobustapproachwasproposedinthesamereferencetoobtaintheThis approach was integrated into our code to obtain the relevant joint variables of the Agile Wrist.the dual equations providing the second joint displacement d2are the dual counterparts of the equations for therobot, namely 13,translational displacement are to be calculated. One approach to solving for 1is to substitute the tan-half identities in the abovetrigonometric equation. This leads to a quadratic equation in tan(1/2), which is then solved using the formula for the roots of thequadratic equation. However, as pointed out in 5, the quadratic-equation approach to the inputoutput analysis of the four-barlinkage leads to polynomial deation, which means in the case at hand that the quadratic equation degenerates into a linearequation,vxs 1s1vys 1c1+ vzc 1c 2=0 9i= i+ ai, for i=1,2,aibeing the perpendicular distance between the ith and (i+1)st joint axes. Moreover, the rstkept as a revolute, 1need not be dualized.above equation has been shown to admit two solutions for angle 15; for the two other joints, a rotational as well as awiththeir dualkinematicsmotionleg arethe base2.3. JacobianTheof thein which,where15. Therepresentthemovingrevolutes339K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343on the robust algorithm proposed in 5 to solve the inverse kinematics of a RCCC closed chain, the robust inverseoftheAgileWristisstraightforward:Arandomniteposeofthemovingplateisspeciedbymeansofarandomscrewinvolving a “small” translation, commensurate with the anticipated manufacturing errors. Now, the joint variables of eachcomputed independently for each leg.To this end, weregard the moving plate in its displaced pose as being rigidlyxedtoplate, thereby forming a closed kinematic chain of the RCCC type.matricesinstantaneous-kinematics equations of the Agile Wrist with legs of the RCC-type are obtained by dualizing the equationscorresponding spherical 3RRR mechanism 2:A +B =0 17A =v1C2 w1Tv2C2 w2Tv3C2 w3T264375;B = diag u1 w1v1;u2 w2v2;u3 w3v3 is the real vector of actuated joint rates, while = vectRRTC16C17is the dual angular velocity vector of the mobile platformdual unit vectors ui, vi, and wi, for i=1,2,3, depend on the architecture and the posture of the robot. These vectorsthe axes of the C joints of the ith leg, with uiand vipertaining to the attachment joints of the ith leg with the base- andplates,respectively, wibeingassociatedwiththeintermediatejointofthesameleg.Moreover,allactuatedjointsbeing, their associated variables and time-derivatives are real numbers and hence, vector should not bear a hat.where3= 3+ C15d316cw1= r11c1+ r21s116dw2= c 1r11s1r21c1C0C1+ s 1r3116ew3= s 1r11s1r21c1C0C1+ c 1r3116fOnce again, Eqs. (16a) & (16b) expand to form four equations, two primal and two dual, thus determining both the rotation 3andthetranslationd3oftheendjoint.TheprimalpartsofEqs.(11a)&(11b)andEqs.(16a)&(16b),aredisplayedcorrespondinglyin Eqs. (10a) & (10b) and Eqs. (15a) & (15b), while their dual parts are included in Appendix A.The foregoing algorithm allows, for a xed pose of the moving platform, the calculation of the input rotation angle 1of theactuator joint, the translational displacements d2and d3, and their corresponding angles of rotation, 2and 3.2.2. The robust inverse kinematics of the Agile WristThe inverse-displacement analysis of the open RCC chain is equivalent to the displacement analysis of the closed RCCC chain14.Thelatterisalsoknownasthespatialfour-barlinkage.Indeed,aposeoftheEEoftheopenchaincanbeprescribedbymeansofanitescrewinvolvinga lineand apitch p. TheEE isthus regarded ashavingattained itsprescribedpose froma referenceposeupon sliding along while concurrently rotating about , rotation and sliding u being related by the pitch: u=p. Thus, constitutes a fourth line in Fig. 1b, along which a C-joint is placed to couple the EE with the robot base, thereby closing the loop.Basedw1= r11c1+