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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 2K6a r t i c l ei n f oa b s t r a c tArticle 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 suitablydimensioned. 2010 Elsevier Ltd. All rights reserved.Keywords:Agile WristSpherical parallel robotsStatistical analysisDual numbersPrinciple of TransferenceInverse-kinematics analysis1. IntroductionSpherical parallel robots are used to orient a rigid body in three-dimensional space. Applications include orienting machine-tool beds and workpieces 1,2, as well as orienting a camera tracking fast-moving objects. The latter application led to thedevelopment of the Agile Eye 3,4.Ideally, all the joints of a spherical mechanism are revolutes, with their axes intersecting at a common point. However,due to machining errors, it is not possible to manufacture such a mechanism with conventional machining operations; high-precision machining is, of course, an option, if one is willing to pay for it. As an alternative, extra degrees of freedom areusually included, to allow for the inevitable machining errors. The designers of the Agile Eye provided the extra degrees offreedom by replacing the passive revolute joints with self-aligning pin joints, which is equivalent to replacing the passiverevolutes 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 offreedom which are necessary and sufficient. Hence, a displacement analysis of the mechanism is needed to determine not only therotations of the revolute joints but also the translations at the unactuated cylindrical joints, in the presence of machining andassembly errors that render the revolute axes non-concurrent. Obviously, the resulting robot will not be spherical anymore, butrather spatial, capable of producing almost spherical displacements.Mechanism and Machine Theory 46 (2011) 335343 Corresponding author. Tel.: +1 514 3984488; fax: +1 514 398 7348.E-mail addresses: alwidyan.jo (K. Al-Widyan), xqmacim.mcgill.ca (X.Q. Ma), angelescim.mcguill.ca (J. Angeles).1Assembling such a closed kinematic chain with manufacturing errors is possible if the links are deformed when assembling the chain.0094-114X/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2010.11.002Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: /locate/mechmtIn 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 analysis of the linkage under study boils down to solving a system of trigonometric equations. The established approach 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 to finding 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.Replacing revolute joints by their cylindrical counterparts results in changing the topology of the robot 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), depicted in Fig. 1a, is a three-degree-of-freedom robot designed as the terminal module of an 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 thespherical 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 firmlyhold 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 finding the passive-joint displacements upon considering a fixed 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.336K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343relations 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 as e3?1= vx vy vz?T1where the subscript 1 refers to the base frame. We recall here that, while a real unit vector represents a direction in Euclideanspace, a dual unit vector represents a line in the same space.Thedisplacement analysisofthesphericalrobotof Fig.1b isoutlinedbelow. Tothis end,the DenavitHartenberg(DH) notation12 is followed, with coordinate frame Fi, for i=1,2,3,4, fixed to the (i1)st link, where 0 denotes the base and 3 the end-effector (EE) of the robot depicted in the foregoing figure. Moreover, a vector v represented in Fiis denoted v ?i. In the inverse-displacement problem (IDP), the attitude of the EE is given by a rotation matrix R in Fi, the joint angles i13that will carry the EEfrom a reference attitude to R being sought. Moreover, let Qi?idenote the rotation matrix that carries vector components in Fi + 1into its counterparts in Fi. We have, moreover, within this notation, ei?ie = 0;0;1?T. For quick reference, the structure of theQi?imatrices is recalled:Qi Qi?icosiisiniisinisiniicosiicosi0ii24352where icosiand isini, with 1and 2shown in Fig. 1b, while 3is the angle made by the axis of the third revolute, parallelto vector e3, with Z4, the Z-axis of F4, attachedto the EE. This axis is user-defined, its sole conditionbeing thatit passes throughthecenter of the wrist. It is not shown to avoid overloading the figure.The key step in solving the IDP of the spherical wrist at hand is setting up the relationeT2e3= cos23whichis derivedby inspectionof Fig.1b.Hence,a coordinate transformationis neededin order toexpressthetwovectorsinvolvedin Eq. (3) in the same coordinate frame, namely, F1, as explained below.e2?1= Q1?1e1?1= 1sin11cos 11?T4which is nothing but the third column of Q1?1. Moreover,e3?1= Q1?1Q2?2e3?35On the other hand,Q1?1Q2?2Q3?3= Q?1whenceQ1?1Q2?2= Q?1QT3hi36and hence,e3?1= Q?1QT3hi3e3?3|zo3where o3denotes the third row of Q3?3. If qijdenotes the (i,j) entry of Q?1, then,e3?1= q123+ q133q223+ q233q323+ q333?T vxvyvz?T7Upon substitution of Eqs. (4) and (7) into Eq. (3), one scalar equation is obtained for 1:vxs1s1vys1c1+ vzc1c2= 08with c()cos() and s()sin(). The derivation of this equation and others needed in this context is described in detail in 13.2Dualizing is sometimes referred to as “putting hats” on all the variables and parameters in the spherical-linkage equations.337K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343Upon dualizing the foregoing expressions, we obtain the counterpart equation for the RCCC chain, namely, vxs 1s1 vys 1c1+ vzc 1c 2= 09where i= i+ ai, for i=1,2, aibeing the perpendicular distance between the ith and (i+1)st joint axes. Moreover, the firstjoint being kept as a revolute, 1need not be dualized.The above equation has been shown to admit two solutions for angle 15; for the two other joints, a rotational as well as atranslational 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 deflation, which means in the case at hand that the quadratic equation degenerates into a linearequation, in the presence of a root close to . As an alternative, a robust approachwas proposed in the same reference to obtain thetwo solutions. This approach was integrated into our code to obtain the relevant joint variables of the Agile Wrist.Further, the dual equations providing the second joint displacement d2are the dual counterparts of the equations for thespherical robot, namely 13,s2s2= s3r12+ c3r13c1+ s3r22+ c3r23s110as2c2= c1s3r12+ c3r13s1c1s3r22+ c3r23c1s1s3r32+ c3r3310bwhere rij, for i,j=1,2,3, are the entries of the rotation matrix R defining the orientation of the end-effector; the dual counterpartsof the above equations ares 2s2= s3 r12+ c 3 r13c1+ s 3 r22+ c 3 r23s111as 2c2= c 1s 3 r12+ c 3 r13s1c 1s 3 r22+ c 3 r23c1s 1s 3 r32+ c 3 r3311bwherei= i+ ?di, with didenoting the signed distance between consecutive normals of the joint axes, according with theDenavitHartenberg notation 13, while rij, for i,j=1,2,3, are the entries of the dual rotation matrixR= R + ?DR12carrying the end-effector from its reference pose to its current pose. The primal part R ofR is a proper orthogonal matrix, while itsdual part involves matrix D, which is the cross-product matrix3of the translation vector d. Moreover, matrixR is a dual properorthogonal matrixdet(R)=+1. The reference attitude is defined by R = 133and D = O33, where 133and O33represent the33 identity and zero matrices, respectively.Matrix R is specified by an angle of rotation and an axis of rotation, given by the unit vector e. This matrix is then foundas 13R = eeT+ cos 133eeT?+ sin E13where E is the cross-product matrix of e, thus obtaining the primal part of matrixR. Hence, each entry of the dual rotation matrix15 is a dual number, namely, rij= rij+ ?ij14ijbeing 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,c3= w1c2+ w2s215as3= w1c2s2+ w2c2c2+ w3s215b3The cross product matrix (CPM) D of the three-dimensional vector d is defined, for any vector v of the same dimension, as D = CPM d d v=v.338K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343withw1= r11c1+ r21s115cw2= c1r11s1r21c1 + s1r3115dw3= s1r11s1r21c1 + c1r3115etheir dual counterparts beingc3=w1c2+w2s216as3= w1c2s2+ w2c2c2+ w3s 216bwhere3= 3+ ?d316cw1= r11c1+ r21s116d w2= c 1 r11s1 r21c1?+ s 1 r3116e w3= s 1 r11s1 r21c1?+ c 1 r3116fOnce again, Eqs. (16a) & (16b) expand to form four equations, two primal and two dual, thus determining both the rotation 3andthe translationd3of the end joint. Theprimal parts of Eqs. (11a) & (11b) andEqs. (16a) & (16b), are displayedcorrespondinglyin Eqs. (10a) & (10b) and Eqs. (15a) & (15b), while their dual parts are included in Appendix A.The foregoing algorithm allows, for a fixed 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.The latteris alsoknownasthespatialfour-bar linkage.Indeed,a poseoftheEEoftheopenchaincanbe prescribedby meansofa finite screw involving a line and a pitch p. The EE is thus regarded as having attained its prescribed pose from a reference poseupon 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.Based on the robust algorithm proposed in 5 to solve the inverse kinematics of a RCCC closed chain, the robust inversekinematics of the Agile Wrist is straightforward: A random finite pose of the moving plate is specified by means of a random screwmotion involving a “small” translation, commensurate with the anticipated manufacturing errors. Now, the joint variables of eachleg are computed independently for each leg. To this end, we regard the moving plate in its displaced pose as being rigidly fixed tothe base plate, thereby forming a closed kinematic chain of the RCCC type.2.3. Jacobian matricesThe instantaneous-kinematics equations of the Agile Wrist with legs of the RCC-type are obtained by dualizing the equationsof the corresponding spherical 3RRR mechanism 2:A +B = 017in which,A = v1? w1T v2? w2T v3? w3T264375;B = diag u1 w1 v1; u2 w2 v2; u3 w3 v3where is the real vector of actuated joint rates, while = vectRRT?is the dual angular velocity vector of the mobile platform15. The dual unit vectors ui, vi, and wi, for i=1,2,3, depend on the architecture and the posture of the robot. These vectorsrepresent the axes of the C joints of the ith leg, with uiand vipertaining to the attachment joints of the ith leg with the base- andthe moving plates, respectively, wibeing associated withthe intermediate joint of the same leg. Moreover, all actuated jointsbeingrevolutes, their associated variables and time-derivatives are real numbers and hence, vector should not bear a hat.339K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 3353432.4. The condition number of a dual matrixThe statistical approach we adopt to determine the minimum space to be allocated for the translations of the C joints of therobot requires that we specify a set of random screw motions 13 of the moving platform. When doing this, the specific set ofrandom numbers selected may not produce a feasible pose of the moving plate, which can be detected by means of the kinematicanalysis of spatial four-bar linkages introduced in 5.Moreover, even in the presence of a feasible pose, this may be attained at an ill-conditioned robot posture, which may lead tojoint translations of various orders of magnitude that of the translation of the input screw motion. Ill-conditioning can be detectedby means of the concept of matrix condition number 16, as applied to the Jacobian matrices of the robot at hand. Thus, postureswith condition numbers larger than a prescribed bound Mwould be avoided either at the task-planing stage or online, by meansof joint-encoder readouts and a suitable algorithm. Henceforth, we assume that the likelihood of occurrence of a posture isinversely proportional to the posture condition number, given in turn by the condition number of the corresponding Jacobianmatrix, as described below.In order to weight the influence of a MP pose in terms of the distance of the corresponding wrist posture from singularity, wehave to evaluate the condition number of a dual matrix, which is the subject of the discussion below.LetM = M + ?M0be a nndual matrix. The inverse ofM, notedM1, takes the form 15:M1= M1?M1M0M118From the above equation, it is apparent that the invertibility of the Jacobian matricesA andB of Eq. (17) depends only on thatof their primal parts. This means that, even, if the dual part of a dual matrix is singular, the dual matrix is still invertible, as long asits primal part is. Hence, the condition number ofM, noted M?, can be taken as that of its primal part, i.e.,M?= MIn our case, only B is needed because it is the Jacobian that need be inverted in the inverse-kinematics analysis. Indeed, fromEq. (17), the analysis calls forB1in order to obtain =B1A.The cylindrical-joint translations are thus weighted using the inverse of the condition number of the primal part ofB as theweighting factor. Hence, each translation value obtained is multiplied by the reciprocal of the corresponding condition number ofB, in order to obtain the weighted-average translation value, i.e.,?di= Nj=11 Bj?dij;i = 2;319where dijis the computed value of the translationof the ith C joint of each leg at the jth random pose of the moving platform, whileN is a statistically significant number of random poses of the moving platform. Moreover,the corresponding standard deviations iof dijj=1N, for i=2,3, are computed asi=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N1Nj=11 Bj?dij?di01A2vuuut;i = 2;3203. ResultsThe code implementing the foregoing algorithms was run for 10,000 random poses of the moving platform. Angles ianddistances ai, for i=1,2, were given as /2+iand ai, respectively, wherei =360;= 360?;ai 3 mm;3 mm?The weighted-mean value and the standard deviation of the results obtained were calculated as summarized in Table 1. As thevalues were weighted with the inverse of the condition number B , a posture with a large condition number does not contributegreatly to the statistical analysis at hand, the contribution of singular postures, with infinitely large condition numbers, being zero.Table 1Cylindrical joint displacements for 10,000 runs (all lengths in mm).Intermediate jointDistal jointMean ()4.79001031.4900103Standard deviation ()0.72000.70003;+32.1552;2.16482.0985;2.1015340K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343Moreover, in order to visualize the output of the displacements of these joints for random poses of the platform, two graphswere plotted, one for each cylindrical joint, as displayed in Figs. 2a and b. In these plots we assumed offset values d1and d4of0.3 mm and twist angles 1of 90.5.From each of these graphs, it is apparent that a large percentage of the displacements lies near a value of zero, the weightedmean values for each joint lying closeto zero as well. The questionnowis: what value of the displacement shouldbe taken in orderto satisfythe majorityof the wrist motionswithoutcausing joint-jamming?The answerto this questioncan be foundby assuming,based on the above-mentioned graphical results, that the distributions of the resulting random displacements are normal, of mean and standard deviation . Hence, deviations below from occur 68% of the time; below 2 occur 95%; and below 3 99% of thetime. The range L=+3 on each side of the joint is taken in order to satisfy 99% of the Agile Wrist poses. Indeed, the solution forthe design of the proximal joint isd2= 4:79 103+ 3 0:72 = 2:1648 mmSimilarly, the design of the distal joint takes the valued3= 1:49 103+ 3 0:70 = 2:1015 mmThe C joints of the Agile Wrist were designed so as to allow for translations of 4.4 mm to be on the safe side. The physicalprototype of the Agile Wrist is shown in Fig. 3, which features simpler, although slightly heavier links than those of Fig. 1a. Inthis figure, curved links with circular cross sections of variable radius are displayed. In the prototype of Fig. 3, curved links withrectangular cross sections of constant dimensions were used, while keeping the central curve of the links of Fig. 1a. The linkswere simplified for purely budgetary reasons. To give an idea of the dimensions of the prototype of Fig. 3, the distance of thewrist center to the rear ends of the motors is 275 mm, its load-carrying capacity is 50 N and its three motors are rated at364 Watt each.4. ConclusionsDesign guidelines were given for achieving robustness against manufacturing and assembly errors in spherical wrists, where astatistical approach was followed to find the optimum clearances of the unactuated C joints. A crucial step in this paper is thedualization of the spherical-linkage kinematic relations in order to allow for the analysis of their corresponding spatialcounterparts, in which the joint axes do not intersect. A procedure was introduced for solving the inverse kinematics of RCC serialrobots by application of the Principle of Transference. The instantaneous-kinematics equations of 3RCC parallel robots weredeveloped by resorting to dual algebra. The condition number of dual matrices was derived, in order to weight the random end-effector poses with the inverse of the condition number of the matrixB that needs to be inverted. It was shown thatB1involvesonly the inverse of its primal part B, for which reason B?= B . A methodology for the robust design and analysis of sphericalrobots was developed and then applied to the determination of the maximum excursions of the cylindrical joints of the AgileWrist, in order to properly dimension the cylindrical joints of the spatial robot thus resulting.642024605001000150020002500642024605001000150020002500abFig. 2. Number of occurrences of random displacements vs. weighted displacement values (mm) for: (a) intermediate joint; (b) distal joint.341K. Al-Widyan et al. / Mechanism and Machine Theory 46 (2011) 335343AcknowledgmentsThe work reported here was partly supported by NSERC (Canadas Natural Sciences and Engineering Research Council) underStrategic Project 215729-98. The partial support provided by CDEN (Canadian Design Engineering Network)is also acknowledged.We acknowledge the work of all those who contributed to earlier versions of the paper: David Bellitto; David Daney; and BrunoMonsarrat.Appendix AThe dual parts of Eqs. (11a) & (11b) ares2d2c2+ a2c2s2= s312a3c3r12c313+ a3s3r13c1 s322a3c3r22c323+ a3s3r23s121s2d2s2+ a2c2c2= s1c1s312c1a3c3r12c1c313+ c1a3s3r13+ a1s1s3r12+ a1s1c3r13+ c1c1s322c1a3c3r22c1c323+ c1a3s3r23+ a1s1s3r22+ a1s1c3r23s1s332s1a3c3r32s1c333+ s1a3s3r33a1c1s3r32a1c1c3r3322The dual parts of Eqs. (16a) & (16b) ared3s3= d2s2r11c1d2s2r21s1+ c211c1+ c221s1+ d2c2s1r31d2c2c1r11s1+ d2c2c1r21c1+ s2s131+ s2a1c1r31s2c111s1+ s2c121c1+ s2a1s1r11s1s2a1