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Robotics and Computer-Integrated Manufacturing 21 (2005) 368378Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrainedconstraint status, a workpiece under any locating scheme falls into one of the following three categories:locating problem using screw theory in 1989. It is concluded that the locating wrenches matrix needs to be full rank toachieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. 3 consideredARTICLE IN PRESS0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.rcim.2004.11.012C3Corresponding author. Tel.: +15088316092; fax: +15088316412.E-mail address: hsong (H. Song).1. Well-constrained (deterministic): The workpiece ismatedat auniqueposition when six locatorsare madeto contactthe workpiece surface.2. Under-constrained: The six degrees of freedom of workpiece are not fully constrained.3. Over-constrained: The six degrees of freedom of workpiece are constrained by more than six locators.In 1985, Asada and By 1 proposed full rank Jacobian matrix of constraint equations as a criterion and formed thebasis of analytical investigations for deterministic locating that followed. Chou et al. 2 formulated the deterministic1. IntroductionA xture is a mechanism used in manufacturing operations to hold a workpiece rmly in position. Being a crucialstep in process planning for machining parts, xture design needs to ensure the positional accuracy and dimensionalaccuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a locationscheme. V-block and pin-hole locating principles are also commonly used.Alocationschemeforamachiningxturemustsatisfyanumberofrequirements.Themostbasicrequirementisthatit must provide deterministic location for the workpiece 1. This notion states that a locator scheme producesdeterministic location when the workpiece cannot move without losing contact with at least one locator. This has beenone of the most fundamental guidelines for xture design and studied by many researchers. Concerning geometryAbstractGeometry constraint is one of the most important considerations in xture design. Analytical formulation of deterministiclocation has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process ofactual xture design practice has not been thoroughly studied. In this paper, a methodology to characterize xturing systemsgeometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognizedwithgiven locatingscheme.All un-constrainedmotionsofaworkpiece inanunder-constraintstatuscanbeautomaticallyidentied.This assists the designer to improve decit locating scheme and provides guidelines for revision to eventually achieve deterministiclocating.r 2005 Elsevier Ltd. All rights reserved.CAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USAReceived 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004Locating completeness evaluation and revision in xture planH. SongC3, Y. R/locate/rcimlocatorworkpiece contact area effects instead of applying point contact. They introduced a contact matrix andpointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson 4suggested that a linear approximation may not be sufcient for some applications such as non-prismatic surfaces ornon-small relative errors.Heproposed asecond-order Taylor expansionwhichalsotakes locatorerror interaction intoaccount. Marin and Ferreira 5 applied Chous formulation on 3-2-1 location and formulated several easy-to-followplanning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyzenon-deterministic location.In the Asada and Bys formulation, they assumed frictionless and point contact between xturing elements andworkpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surfacefunction is gi(as shown in Fig. 1).The surface function isdened as giqC30: To be deterministic, there should be a unique solution for the followingequation set for all locators.giq0; i 1;2; .; n, (1)where n is the number of locators and q x0; y0; z0;y0;f0;c0C138 represents the position and orientation of theworkpiece.Only considering the vicinity of desired location qC3; where q qC3Dq; Asada and By showed thatARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 369giqgiqC3hiDq, (2)where hiis the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locatingrequirement can be satised if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solutionq qC3:rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0:qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0:qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc026666666664377777777758:9=; 6. (3)Upongivena3-2-1locatingscheme, therankofaJacobianmatrixforconstraintequationstellstheconstraintstatusas shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one freemotion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme hasmore than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that itcan be removed without affecting the geometry constrain status of the workpiece.For locating a model other than 3-2-1, datum frame can be established to extract equivalent locating points. Hu 6has developed a systematic approach for this purpose. Hence, this criterion can be applied to all locating schemes.X Y Z O X Y Z O (x0,y0,z0) gi UCS WCS Workpiece Fig. 1. Fixturing system model.They further introduced several indexes derived from those matrixes to evaluate locator congurations, followed byoptimization through constrained nonlinear programming. Their analytical study, however, does not concern theARTICLE IN PRESSrevision of non-deterministic locating. Currently, there is no systematic study on how to deal with a xture design thatfailed to provide deterministic location.2. LocatingcompletenessevaluationIf deterministic location is not achieved by designed xturing system, it is as important for designers to knowwhat the constraint status is and how to improve the design. If the xturing system is over-constrained, informa-tion about the unnecessary locators is desired. While under-constrained occurs, the knowledge about all the un-constrained motions of a workpiece may guide designers to select additional locators and/or revise the locatingscheme more efciently. A general strategy to characterize geometry constraint status of a locating scheme is describedin Fig. 2.In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status (see Appendixfor derivation of locating matrix). The deterministic locating requires six locators that provide full rank locatingmatrix WL:As shown in Fig. 3, for given locator number n; locating normal vector ai; bi; ciC138 and locating position xi; yi; ziC138 foreach locator, i 1;2; .; n; the n C26 locating matrix can be determined as follows:a1b1c1c1y1C0 b1z1a1z1C0 c1x1b1x1C0 a1y1: : : :2637Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module intheir automated computer-aided xture design verication system. Their CAFDV system can calculate the Jacobianmatrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivityto locating error.Xiong et al. 8 presented an approach to check the rank of locating matrix WL(see Appendix). They also intro-duced left/right generalized inverse of the locating matrix to analyze the geometric errors of workpiece. It hasbeen shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators arerelated as follows:Well-constrained : DX WLDr, (4)Over-constrained : DX WTLWLC01WTLDr, (5)Under-constrained : DX WTLWLWTLC01Dr I6C26C0 WTLWLWTLC01WLl, (6)where l is an arbitrary vector.Table 1Rank Number of locators Statuso 6 Under-constrained 6 6 Well-constrained 6 46 Over-constrainedH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370WLaibiciciyiC0 biziaiziC0 cixibixiC0 aiyi: : : :anbncncnynC0 bnznanznC0 cnxnbnxnC0 anyn666664777775.(7)When rankWL6 and n 6; the workpiece is well-constrained.When rankWL6 and n46; the workpiece is over-constrained. This means there are n C06 unnecessary locatorsin the locating scheme. The workpiece will be well-constrained without the presence of those n C06 locators. Themathematical representationforthisstatusisthat thereare n C06 rowvectorsinlocating matrix thatcanbeexpressedas linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist oneARTICLE IN PRESSlocatdeterm.be3.workpiH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 371ing scheme that provides deterministic location. The developed algorithm uses the following approach toine the unnecessary locators:Find all the combination of n C06 locators.For each combination, remove that n C06 locators from locating scheme.Recalculate the rank of locating matrix for the left six locators.If the rank remains unchanged, the removed n C06 locators are responsible for over-constrained status.This method may yield multi-solutions and require designer to determine which set of unnecessary locators shouldremoved for the best locating performance.When rankWLo6; the workpiece is under-constrained.AlgorithmdevelopmentandimplementationThe algorithm to be developed here will dedicate to provide information on un-constrained motions of theece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/Fig. 2. Geometry constraint status characterization.X Z Y (a1,b1,c1) 2,b2,c2) (x1,y1,z1) (x2,y2,z2) (ai,bi,ci) (xi,yi,zi) (aFig. 3. A simplied locating scheme.orientijLLLARTICLE IN PRESS3725.To identify allthe un-constrained motions oftheworkpiece, V dxi;dyi;dzi;daxi;dayi;daziC138 isintroducedsuchthatV DX 0. (9)Since rankDXo6; there must exist non-zero V that satises Eq. (9). Each non-zero solution of V represents an un-constrained motion. Each term of V represents a component of that motion. For example, 0;0;0;3;0;0C138 says that therotation about x-axisisnotconstrained. 0;1;1;0;0;0C138 meansthat theworkpiececanmovealongthedirection given byvector 0;1;1C138: There could be innite solutions. The solution space, however, can be constructed by 6C0 rankWLbasic solutions. Following analysis is dedicated to nd out the basic solutions.From Eqs. (8) and (9)VX dxDx dyDy dzDz daxDaxdayDaydazDaz dxXni1w1iDridyXni1w2iDridzXni1w3iDridaxXni1w4iDridayXni1w5iDridazXni1w6iDriXni1Vw1i; w2i; w3i; w4i; w5i; w6iC138TDri 0. 10Eq. (10) holds for 8Driif and only if Eq. (11) is true for 8i1pipn:Vw1i; w2i; w3i; w4i; w5i; w6iC138T 0. (11)Eq. (11) illustrates the dependency relationships among row vectors of Wr: In special cases, say, all w1jequal to zero,V has an obvious solution 1, 0, 0, 0, 0, 0, indicating displacement along the x-axis is not constrained. This is easy tounderstand because Dx 0 in this case, implying that the corresponding position error of the workpiece is notdependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combinedmotion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. Forinstance, 9w1ja0;w2ja0; w1jC0w2jfor 8j: Inthisscenario,theworkpiece cannotmovealong x-ory-axis.However,itcan move along the diagonal line between x-andy-axis dened by vector 1, 1, 0.To nd solutions for general cases, the following strategy was developed:1. Eliminate dependent row(s) from locating matrix. Let r rank WL; n number of locator. If ron; create a vectorin n C0 r dimension space U u1: uj: unC0rhi1pjpn C0 r; 1pujpn: Select ujin the way that rankWLr still holds after setting all the terms of all the ujth row(s) equal to zero. Set r C26 modied locating matrixWLMa1b1c1c1y1C0 b1z1a1z1C0 c1x1b1x1C0 a1y1: : : :aibiciciyiC0 biziaiziC0 cixibixiC0 aiyi: : : :anbncncnynC0 bnznanznC0 cnxnbnxnC0 anyn2666666437777775rC26,whergeometation errors and locator errors can be expressed as follows:DX DxDyDzaxayaz2666666666437777777775w11: w1i: w1nw21: w2i: w2nw31: w3i: w3nw41: w4i: w4nw51: w5i: w5nw61: w6i: w6n2666666666437777777775C1Dr1:Dri:Drn2666666437777775, (8)e Dx;Dy;Dz;ax;ay;azare displacement along x, y, z axis and rotation about x, y, z axis, respectively. Driisric error of the ith locator. w is dened by right generalized inverse of the locating matrix Wr WTW WTC01H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378where i 1;2; :; niauj:4.6.constrExamplvectorARTICLE IN PRESSL3: 0, 0, 10, 2, 1, 00,L4: 0, 1, 00, 3, 0, 20,L5: 0, 1, 00, 1, 0, 20.Consequently, the locating matrix is determined.WL001 3 C010001 3 C030001 1 C020010C02032666666437777775.LLvs:v666647775wqki:wqkr66647775C1wl1: wli: wlr:w61: w6i: w6r66647775,where s 1;2; :;6saqj; saqk; l 1;2; :;6 laqj:Repeat step 4 (select another term from Q) and step 5 until all 6C0 r basic solutions have been determined.Based on this algorithm, a C+ program was developed to identify the under-constrained status and un-ained motions.e1. In a surface grinding operation, a workpiece is located on a xture system as shown in Fig. 4. The normaland position of each locator are as follows:1: 0, 0, 10, 1, 3, 00,2: 0, 0, 10,3,3,00,Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found asfollows.v1:26663777wqk1:26663777w11: w1i: w1r:26663777C015.Wrmwl1: wli: wlr:w61: w6i: w6r666477756C26,where l 1;2; :;6 laqj:Normalize the free motion space. Suppose V V1; V2; V3; V4; V5; V6C138 is one of the basic solutions of Eq. (10) withall six terms undetermined. Select a term qkfrom vector Q1pkp6C0 r: SetVqkC01;Vqj 0 j 1;2; :;6C0 r; jak;(2. Compute the 6C2 n right generalized inverse of the modied locating matrixWr WTLMWLMWTLMC01w11: w1i: w1rw21: w2i: w2rw31: w3i: w3rw41: w4i: w4rw51: w5i: w5rw61: w6i: w6r26666666664377777777756C2r3. Trim Wrdown to a r C2 rfull rank matrix Wrm: r rankWLo6: Construct a 6C0 r dimension vector Q q1: qj: q6C0rhi1pjp6C0 r; 1pqjpn: Select qjin the way that rankWrr still holds after setting all theterms of all the qjth row(s) equal to zero. Set r C2 r modied inverse matrixw11: w1i: w1r:26663777H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 373010C0201ARTICLE IN PRESSThis locating system provides under-constrained positioning since rankWL5o6: The program then calculatesthe right generalized inverse of the locating matrix.Wr00 0000:50:5 C01 C00:51:50:75 C01:25 1:50 00:25 0:25 C00:50 00:5 C00:50000000:5 C00:526666666643777777775.The rst row is recognized as a dependent row because removal of this row does not affect rank of the matrix. Theother ve rows are independent rows. A linear combination of the independent rows is found according therequirementinstep5oftheprocedureforunder-constrainedstatus.Thesolutionforthisspecialcaseisobviousthatallthe coefcients are zero. Hence, the un-constrained motion of workpiece can be determined as V C01; 0; 0; 0; 0; 0C138:This indicates that the workpiece can move along x direction. Based on this result, an additional locator should beemployed to constraint displacement of workpiece along x-axis.XZYL3L4L5L2L1Fig. 4. Under-constrained locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374Example2. Fig. 5 shows a knuckle with 3-2-1 locating system. The normal vector and position of each locator in thisinitial design are as follows:L1: 0, 1, 00, 896, C0877, C05150,L2: 0, 1, 00, 1060, C0875, C03780,L3: 0, 1, 00, 1010, C0959, C06120,L4: 0.9955, C00.0349, 0.0880, 977, C0902, C06240,L5: 0.9955, C00.0349, 0.0880, 977, C0866, C06240,L6: 0.088, 0.017, C00.9960, 1034, C0864, C03590.The locating matrix of this conguration isWL0 1 0 515:000:896001 0378: 1:06000 1 0 612:00:01000:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:86380:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:82800:0880 0:0170 C00:9960 866:6257998 :2466 0:093626666666643777777775,rankWL5o6 reveals that the workpiece is under-constrained. It is found that one of the rst ve rows can beremoved without varying the rank of locating matrix. Suppose the rst row, i.e., locator L1is removed from WL; theARTICLE IN PRESSmodied locating matrix turns intoWLM010378:001:06000 1 0 612: :01000:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:86380:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:82800:0880 0:0170 C00:996 866:6257998 :2466 0:09362666666437777775.The right generalized inverse of the modied locating matrix isWr1:8768 C01:8607 C020:6665 21:3716 0:49953:0551 C02:0551 C032:4448 32:4448 0C01:0956 1:0862 12:0648 C012:4764 C00:2916C00:0044 0:0044 0:0061 C00:0061 00:0025 C00:0025 0:0065 C00:0069 0:0007C00:0004 0:0004 0:0284 C00:0284 026666666643777777775.The program checked the dependent row and found every row is dependent on other ve rows. Without losinggenerality, the rst row is regarded as dependent row. The 5C25 modied inverse matrix is2 3Fig. 5. Knuckle 610 (modied from real design).H. Song, Y. Ron
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