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Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained constraint 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 to achieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. 3 considered ARTICLE IN PRESS 0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2004.11.012 C3 Corresponding author. Tel.: +15088316092; fax: +15088316412. E-mail address: hsongwpi.edu (H. Song). 1. Well-constrained (deterministic): The workpiece ismatedat auniqueposition when six locatorsare madeto contact the 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 the basis of analytical investigations for deterministic locating that followed. Chou et al. 2 formulated the deterministic 1. Introduction A xture is a mechanism used in manufacturing operations to hold a workpiece rmly in position. Being a crucial step in process planning for machining parts, xture design needs to ensure the positional accuracy and dimensional accuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a location scheme. V-block and pin-hole locating principles are also commonly used. Alocationschemeforamachiningxturemustsatisfyanumberofrequirements.Themostbasicrequirementisthat it must provide deterministic location for the workpiece 1. This notion states that a locator scheme produces deterministic location when the workpiece cannot move without losing contact with at least one locator. This has been one of the most fundamental guidelines for xture design and studied by many researchers. Concerning geometry Abstract Geometry constraint is one of the most important considerations in xture design. Analytical formulation of deterministic location has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process of actual xture design practice has not been thoroughly studied. In this paper, a methodology to characterize xturing systems geometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognized withgiven locatingscheme.All un-constrainedmotionsofaworkpiece inanunder-constraintstatuscanbeautomaticallyidentied. This assists the designer to improve decit locating scheme and provides guidelines for revision to eventually achieve deterministic locating. r 2005 Elsevier Ltd. All rights reserved. CAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA Received 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004 Locating completeness evaluation and revision in xture plan H. Song C3 , Y. Rong locatorworkpiece contact area effects instead of applying point contact. They introduced a contact matrix and pointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson 4 suggested that a linear approximation may not be sufcient for some applications such as non-prismatic surfaces or non-small relative errors.Heproposed asecond-order Taylor expansionwhichalsotakes locatorerror interaction into account. Marin and Ferreira 5 applied Chous formulation on 3-2-1 location and formulated several easy-to-follow planning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyze non-deterministic location. In the Asada and Bys formulation, they assumed frictionless and point contact between xturing elements and workpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surface function is g i (as shown in Fig. 1). The surface function isdened as g i q C3 0: To be deterministic, there should be a unique solution for the following equation set for all locators. g i q0; i 1;2; .; n, (1) where n is the number of locators and q x 0 ; y 0 ; z 0 ;y 0 ;f 0 ;c 0 C138 represents the position and orientation of the workpiece. Only considering the vicinity of desired location q C3 ; where q q C3 Dq; Asada and By showed that ARTICLE IN PRESS H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 369 g i qg i q C3 h i Dq, (2) where h i is the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locating requirement can be satised if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solution q q C3 : rank qg 1 qx 0 qg 1 qy 0 qg 1 qz 0 qg 1 qy 0 qg 1 qf 0 qg 1 qc 0 : qg i qx 0 qg i qy 0 qg i qz 0 qg i qy 0 qg i qf 0 qg i qc 0 : qg n qx 0 qg n qy 0 qg n qz 0 qg n qy 0 qg n qf 0 qg n qc 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 8 : 9 = ; 6. (3) Upongivena3-2-1locatingscheme, therankofaJacobianmatrixforconstraintequationstellstheconstraintstatus as shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one free motion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme has more than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that it can 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 6 has 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 (x 0 ,y 0 ,z 0 ) g i UCS WCS Workpiece Fig. 1. Fixturing system model. They further introduced several indexes derived from those matrixes to evaluate locator congurations, followed by optimization through constrained nonlinear programming. Their analytical study, however, does not concern the ARTICLE IN PRESS revision of non-deterministic locating. Currently, there is no systematic study on how to deal with a xture design that failed to provide deterministic location. 2. Locatingcompletenessevaluation If deterministic location is not achieved by designed xturing system, it is as important for designers to know what 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 locating scheme more efciently. A general strategy to characterize geometry constraint status of a locating scheme is described in Fig. 2. In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status (see Appendix for derivation of locating matrix). The deterministic locating requires six locators that provide full rank locating matrix W L : As shown in Fig. 3, for given locator number n; locating normal vector a i ; b i ; c i C138 and locating position x i ; y i ; z i C138 for each locator, i 1;2; .; n; the n C26 locating matrix can be determined as follows: a 1 b 1 c 1 c 1 y 1 C0 b 1 z 1 a 1 z 1 C0 c 1 x 1 b 1 x 1 C0 a 1 y 1 : : : : 2 6 3 7 Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module in their automated computer-aided xture design verication system. Their CAFDV system can calculate the Jacobian matrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivity to locating error. Xiong et al. 8 presented an approach to check the rank of locating matrix W L (see Appendix). They also intro- duced left/right generalized inverse of the locating matrix to analyze the geometric errors of workpiece. It has been shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators are related as follows: Well-constrained : DX W L Dr, (4) Over-constrained : DX W T L W L C01 W T L Dr, (5) Under-constrained : DX W T L W L W T L C01 Dr I 6C26 C0 W T L W L W T L C01 W L l, (6) where l is an arbitrary vector. Table 1 Rank Number of locators Status o 6 Under-constrained 6 6 Well-constrained 6 46 Over-constrained H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370 W L a i b i c i c i y i C0 b i z i a i z i C0 c i x i b i x i C0 a i y i : : : : a n b n c n c n y n C0 b n z n a n z n C0 c n x n b n x n C0 a n y n 6 6 6 6 6 4 7 7 7 7 7 5 .(7) When rankW L 6 and n 6; the workpiece is well-constrained. When rankW L 6 and n46; the workpiece is over-constrained. This means there are n C06 unnecessary locators in the locating scheme. The workpiece will be well-constrained without the presence of those n C06 locators. The mathematical representationforthisstatusisthat thereare n C06 rowvectorsinlocating matrix thatcanbeexpressed as linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist one ARTICLE IN PRESS locat determ 1. 2. 3. 4. be 3. workpi H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 371 ing scheme that provides deterministic location. The developed algorithm uses the following approach to ine 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 should removed for the best locating performance. When rankW L o6; the workpiece is under-constrained. Algorithmdevelopmentandimplementation The algorithm to be developed here will dedicate to provide information on un-constrained motions of the ece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ Fig. 2. Geometry constraint status characterization. X Z Y (a 1 ,b 1 ,c 1 ) 2 ,b 2 ,c 2 ) (x 1 ,y 1 ,z 1 ) (x 2 ,y 2 ,z 2 ) (a i ,b i ,c i ) (x i ,y i ,z i ) (a Fig. 3. A simplied locating scheme. orient ij L L L ARTICLE IN PRESS 372 5. To identify allthe un-constrained motions oftheworkpiece, V dx i ;dy i ;dz i ;da xi ;da yi ;da zi C138 isintroducedsuchthat V 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 the rotation about x-axisisnotconstrained. 0;1;1;0;0;0C138 meansthat theworkpiececanmovealongthedirection given by vector 0;1;1C138: There could be innite solutions. The solution space, however, can be constructed by 6C0 rankW L basic solutions. Following analysis is dedicated to nd out the basic solutions. From Eqs. (8) and (9) VX dxDx dyDy dzDz da x Da x da y Da y da z Da z dx X n i1 w 1i Dr i dy X n i1 w 2i Dr i dz X n i1 w 3i Dr i da x X n i1 w 4i Dr i da y X n i1 w 5i Dr i da z X n i1 w 6i Dr i X n i1 Vw 1i ; w 2i ; w 3i ; w 4i ; w 5i ; w 6i C138 T Dr i 0. 10 Eq. (10) holds for 8Dr i if and only if Eq. (11) is true for 8i1pipn: Vw 1i ; w 2i ; w 3i ; w 4i ; w 5i ; w 6i C138 T 0. (11) Eq. (11) illustrates the dependency relationships among row vectors of W r : In special cases, say, all w 1j equal 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 to understand because Dx 0 in this case, implying that the corresponding position error of the workpiece is not dependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combined motion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. For instance, 9w 1j a0;w 2j a0; w 1j C0w 2j for 8j: Inthisscenario,theworkpiece cannotmovealong x-ory-axis.However,it can 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 W L ; n number of locator. If ron; create a vector in n C0 r dimension space U u 1 : u j : u nC0r hi 1pjpn C0 r; 1pu j pn: Select u j in the way that rankW L r still holds after setting all the terms of all the u j th row(s) equal to zero. Set r C26 modied locating matrix W LM a 1 b 1 c 1 c 1 y 1 C0 b 1 z 1 a 1 z 1 C0 c 1 x 1 b 1 x 1 C0 a 1 y 1 : : : : a i b i c i c i y i C0 b i z i a i z i C0 c i x i b i x i C0 a i y i : : : : a n b n c n c n y n C0 b n z n a n z n C0 c n x n b n x n C0 a n y n 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 rC26 , wher geomet ation errors and locator errors can be expressed as follows: DX Dx Dy Dz a x a y a z 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 w 11 : w 1i : w 1n w 21 : w 2i : w 2n w 31 : w 3i : w 3n w 41 : w 4i : w 4n w 51 : w 5i : w 5n w 61 : w 6i : w 6n 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 C1 Dr 1 : Dr i : Dr n 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 , (8) e Dx;Dy;Dz;a x ;a y ;a z are displacement along x, y, z axis and rotation about x, y, z axis, respectively. Dr i is ric error of the ith locator. w is dened by right generalized inverse of the locating matrix W r W T W W T C01 H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 where i 1;2; :; niau j : 4. 6. constr Exampl vector ARTICLE IN PRESS L 3 : 0, 0, 1 0 , 2, 1, 0 0 , L 4 : 0, 1, 0 0 , 3, 0, 2 0 , L 5 : 0, 1, 0 0 , 1, 0, 2 0 . Consequently, the locating matrix is determined. W L 001 3 C010 001 3 C030 001 1 C020 010C0203 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . L L v s : v 6 6 6 6 4 7 7 7 5 w q k i : w q k r 6 6 6 4 7 7 7 5 C1 w l1 : w li : w lr : w 61 : w 6i : w 6r 6 6 6 4 7 7 7 5 , where s 1;2; :;6saq j ; saq k ; l 1;2; :;6 laq j : 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 normal and position of each locator are as follows: 1 : 0, 0, 1 0 , 1, 3, 0 0 , 2 : 0, 0, 1 0 ,3,3,0 0 , Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found as follows. v 1 : 2 6 6 6 3 7 7 7 w q k 1 : 2 6 6 6 3 7 7 7 w 11 : w 1i : w 1r : 2 6 6 6 3 7 7 7 C01 5. W rm w l1 : w li : w lr : w 61 : w 6i : w 6r 6 6 6 4 7 7 7 5 6C26 , where l 1;2; :;6 laq j : Normalize the free motion space. Suppose V V 1 ; V 2 ; V 3 ; V 4 ; V 5 ; V 6 C138 is one of the basic solutions of Eq. (10) with all six terms undetermined. Select a term q k from vector Q1pkp6C0 r: Set V q k C01; V q j 0 j 1;2; :;6C0 r; jak; ( 2. Compute the 6C2 n right generalized inverse of the modied locating matrix W r W T LM W LM W T LM C01 w 11 : w 1i : w 1r w 21 : w 2i : w 2r w 31 : w 3i : w 3r w 41 : w 4i : w 4r w 51 : w 5i : w 5r w 61 : w 6i : w 6r 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 6C2r 3. Trim W r down to a r C2 rfull rank matrix W rm : r rankW L o6: Construct a 6C0 r dimension vector Q q 1 : q j : q 6C0r hi 1pjp6C0 r; 1pq j pn: Select q j in the way that rankW r r still holds after setting all the terms of all the q j th row(s) equal to zero. Set r C2 r modied inverse matrix w 11 : w 1i : w 1r : 2 6 6 6 3 7 7 7 H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 373 010C0201 ARTICLE IN PRESS This locating system provides under-constrained positioning since rankW L 5o6: The program then calculates the right generalized inverse of the locating matrix. W r 00 000 0:50:5 C01 C00:51:5 0:75 C01:25 1:50 0 0:25 0:25 C00:50 0 0:5 C00:5000 0000:5 C00:5 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The rst row is recognized as a dependent row because removal of this row does not affect rank of the matrix. The other ve rows are independent rows. A linear combination of the independent rows is found according the requirementinstep5oftheprocedureforunder-constrainedstatus.Thesolutionforthisspecialcaseisobviousthatall the 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 be employed to constraint displacement of workpiece along x-axis. X Z Y L 3 L 4 L 5 L 2 L 1 Fig. 4. Under-constrained locating scheme. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374 Example2. Fig. 5 shows a knuckle with 3-2-1 locating system. The normal vector and position of each locator in this initial design are as follows: L 1 : 0, 1, 0 0 , 896, C0877, C0515 0 , L 2 : 0, 1, 0 0 , 1060, C0875, C0378 0 , L 3 : 0, 1, 0 0 , 1010, C0959, C0612 0 , L 4 : 0.9955, C00.0349, 0.088 0 , 977, C0902, C0624 0 , L 5 : 0.9955, C00.0349, 0.088 0 , 977, C0866, C0624 0 , L 6 : 0.088, 0.017, C00.996 0 , 1034, C0864, C0359 0 . The locating matrix of this conguration is W L 0 1 0 515:000:8960 01 0378: 1:0600 0 1 0 612:00:0100 0:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:8638 0:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:8280 0:0880 0:0170 C00:9960 866:6257998 :2466 0:0936 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 , rankW L 5o6 reveals that the workpiece is under-constrained. It is found that one of the rst ve rows can be removed without varying the rank of locating matrix. Suppose the rst row, i.e., locator L 1 is removed from W L ; the ARTICLE IN PRESS modied locating matrix turns into W LM 010378:001:0600 0 1 0 612: :0100 0:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:8638 0:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:8280 0:0880 0:0170 C00:996 866:6257998 :2466 0:0936 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . The right generalized inverse of the modied locating matrix is W r 1:8768 C01:8607 C020:6665 21:3716 0:4995 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The program checked the dependent row and found every row is dependent on other ve rows. Without losing generality, the rst row is regarded as dependent row. The 5C25 modied inverse matrix is 2 3 Fig. 5. Knuckle 610 (modied from real design). H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 375 W rm 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 6 6 6 6 6 6 4 7 7 7 7 7 7 5 . The undetermined solution is V C01; v 2 ; v 3 ; v 4 ; v 5 ; v 6 C138: To calculate the ve undetermined terms of V according to step 5, 1:8768 C01:8607 C020:6665 21:3716 0:4995 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 T C1 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 C01 0; C01:713; C00:0432; C00:0706; 0:04C138. Substituting this result into the undetermined solution yields V C01;0; C01:713; C00:0432; C00:0706; 0:04C138 This vector represents a free motion dened by the combination of a displacement along C01, 0, C01.713 direction combine