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表 2.1.1 机械加工工序卡产品型号零件图号无锡职业技术学 院机械加工工序卡机械加工工序卡产品名称零件名称气门摇臂共 页第 页车间 工序号工序名称材料牌号金工90铣削45毛坯种类毛坯外形尺寸每毛坯可制作数每台件数铸造11设备名称设备型号设备编号同时加工件数铣床XA50321夹具编号夹具名称切削液铣床平口钳工位夹具编号工位器具名称工序工时准终单件工时/min工步号工步名称工艺装备主轴转速(r/min)切削速度(m/min)进给量(mm)背吃刀量(mm)进给次数机动单件描图粗铣 26H7 两端面YT15 硬质合金铣刀77077.40.41110.01151.71精铣 26H7 两端面YT15 硬质合金铣刀1200120.60.30.510.01151.7描校底图号装订号设计 (日期)审核 (日期)标准化 (日期)会签 (日期)标记处数更改文件号签字日期标记处数更改文件号签字日期表 2.1.2 机械加工工序卡产品型号零件图号无锡职业技术学 院机械加工工序卡机械加工工序卡产品名称零件名称气门摇臂共 页第 页车间 工序号工序名称材料牌号金工150铣槽45毛坯种类毛坯外形尺寸每毛坯可制作数每台件数铸造11设备名称设备型号设备编号同时加工件数铣床XA50321夹具编号夹具名称切削液专用夹具工位夹具编号工位器具名称工序工时准终单件工时/min工步号工步名称工艺装备主轴转速(r/min)切削速度(m/min)进给量(mm)背吃刀量(mm)进给次数机动单件描图铣 2mm 油槽D=2 圆柱立铣刀300.190.2210.0090.395描校底图号装订号设计 (日期)审核 (日期)标准化 (日期)会签 (日期)标记处数更改文件号签字日期标记处数更改文件号签字日期表 2.3 机械加工工艺卡产品型号零(部)件图号共 3 页无锡职业技术学院机械加工工艺卡片产品名称零(部)件名称气门摇臂第 1 页材料牌号45毛坯种类铸造毛坯外形尺寸每毛坯件数1每台件数1备注切削用量工艺装备名称及编号工时定额工 序装 夹工步工序内容同时加工零件数背吃刀量(mm)切削 速度m/min每分钟转数进给量(mm)设备名称及编号夹具刀具量具技 术 等 级单件准终101铸造毛坯1201退火13011钻26H7 孔至25。12515.311950.48Z52525麻花钻塞规0.392扩孔25 至25.85。10.8522.082720.62Z52525.85扩孔钻塞规0.313铰孔25.85 至26H7。10.1515.921950.28Z52526 铰刀塞规0.974021粗铣26H7 两端面。1177.47700.41XA5032平口钳D80 端铣刀卡尺1.71编制(日期)审核(日期)会签(日期)产品型号零(部)件图号共 3 页无锡职业技术学院机械加工工艺卡片产品名称零(部)件名称气门摇臂第 2 页材料牌号45毛坯种类铸造毛坯外形尺寸每毛坯件数1每台件数1备注切削用量工艺装备名称及编号工时定额工 序装 夹工步工序内容同时加工零件数背吃刀量(mm)切削 速度m/min每分钟转数进给量(mm)设备名称及编号夹具刀具量具技 术 等 级单件准终2精铣26H7 两端面，并倒角145并保证尺寸 29mm。10.5120.612000.3XA5032平口钳D80 端铣刀卡尺1.75011粗铣16 上端面1128.263750.18XA5032平口钳D80 端铣刀卡尺0.232精铣16 上端面，并保证尺寸17mm。10.5 35.84750.13XA5032平口钳D24 端铣刀卡尺0.286011钻 M10X1-6H 螺纹底孔8.7。18.745.6913600.1Z525平口钳8.7麻花钻0.242在螺纹底孔两端倒角0.545。1Z525平口钳9.7麻 花钻3攻螺纹至要求。1Z525平口钳10丝锥卡尺5编制(日期)审核(日期)会签(日期)产品型号零(部)件图号共 3 页无锡职业技术学院机械加工工艺卡片产品名称零(部)件名称气门摇臂第 3 页材料牌号45毛坯种类铸造毛坯外形尺寸每毛坯件数1每台件数1备注切削用量工艺装备名称及编号工时定额工 序装 夹工步工序内容同时加工零件数背吃刀量(mm)切削 速度m/min每分钟转数进给量(mm)设备名称及编号夹具刀具量具技 术 等 级单件准终7011钻斜台阶油孔2.5、1.6 至要求。12213600.1Z525平口钳2.5、1.6麻花钻1.9768011粗铣 R9.5 圆弧面。113.141000.18XA5032平口钳D25 立铣刀卡尺0.199011对 R9.5 圆弧面高频淬火，保证充足淬火深度。12正火处理。1 10011精磨 R9.5 圆弧面，保证淬火深度，并保证尺寸 10mm。1 0.5158314400.03M7130砂轮卡尺2.0911011铣 2mm 油槽120.19300.2XA5032专用夹具D2 立铣刀卡尺0.395120 11钳工去毛刺，抛光 R9.5 圆弧。1砂纸编制(日期)审核(日期)会签(日期)表 2.2 机械加工工艺过程卡产品型号零件图号机械加工工艺过程卡片产品名称零件名称气门摇臂共 1 页第 1 页材料牌号45毛坯种类模锻件毛坯外形尺寸每毛坯可制作数1每台件数 1备注工序号工序 名称工序内容车间工段设备工艺装备工时/min10铸造熔模铸造毛坯锻工20热处理退火热处理30 钻削钻 26H7 孔至 25金工Z52540钻削扩孔 25 至 25.85金工Z52550钻削绞孔 25.85 至 26H7金工Z52560钻削 钻 M10X1-6H 螺纹底孔金工Z52570铣削粗精铣 16 上端面金工XA5032描图80铣削粗铣 R9.5 圆弧面金工XA503290铣削粗精铣 26H7 上下两端面金工XA5032描校100钻削倒角 1X45金工Z525110钻削钻 2.5、1.6 油孔金工Z525底图号120铣削粗铣 R9.5 圆弧面金工XA5032130热处理对 R9。5 圆弧高频淬火热处理140磨磨 R9.5 圆弧面金工M7130装订号150铣削铣 2mm 油槽金工XA5032设计 (日期)审核 (日期)标准化 (日期)会签 (日期)标记处数更改文件号签字日期标记处数更改文件号签字日期Robotics and Computer-Integrated Manufacturing 21 (2005) 368378Locating completeness evaluation and revision in fixture planH. Song?, Y. RongCAM 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 2004AbstractGeometry constraint is one of the most important considerations in fixture design. Analytical formulation of deterministiclocation has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process ofactual fixture design practice has not been thoroughly studied. In this paper, a methodology to characterize fixturing systemsgeometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognizedwith given locating scheme. All un-constrained motions of a workpiece in an under-constraint status can be automatically identified.This assists the designer to improve deficit locating scheme and provides guidelines for revision to eventually achieve deterministiclocating.r 2005 Elsevier Ltd. All rights reserved.Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained1. IntroductionA fixture is a mechanism used in manufacturing operations to hold a workpiece firmly in position. Being a crucialstep in process planning for machining parts, fixture 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.A location scheme for a machining fixture must satisfy a number of requirements. The most basic requirement is thatit 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 fixture design and studied by many researchers. Concerning geometryconstraint status, a workpiece under any locating scheme falls into one of the following three categories:1. Well-constrained (deterministic): The workpiece is mated at a unique position when six locators are made to 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 deterministiclocating 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 PRESS/locate/rcim0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.rcim.2004.11.012?Corresponding author. Tel.: +15088316092; fax: +15088316412.E-mail address: hsong (H. Song).locatorworkpiece 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 sufficient for some applications such as non-prismatic surfaces ornon-small relative errors. He proposed a second-order Taylor expansion which also takes locator error 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 fixturing 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 is defined as giq? 0: To be deterministic, there should be a unique solution for the followingequation set for all locators.giq 0;i 1;2;.;n,(1)where n is the number of locators and q x0;y0;z0;y0;f0;c0? represents the position and orientation of theworkpiece.Only considering the vicinity of desired location q?; where q q? Dq; Asada and By showed thatgiq giq? hiDq,(2)where hiis the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locatingrequirement can be satisfied if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solutionq q?:rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0:qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0:qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc026666666664377777777758:9=; 6.(3)Upon given a 3-2-1 locating scheme, the rank of a Jacobian matrix for constraint equations tells the constraint statusas 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.ARTICLE IN PRESSX Y Z O X Y Z O (x0,y0,z0) gi UCS WCS Workpiece Fig. 1. Fixturing system model.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378369Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module intheir automated computer-aided fixture design verification 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 WTLWL?1WTLDr,(5)Under-constrained :DX WTLWLWTL?1Dr I6?6? WTLWLWTL?1WLl,(6)where l is an arbitrary vector.They further introduced several indexes derived from those matrixes to evaluate locator configurations, followed byoptimization through constrained nonlinear programming. Their analytical study, however, does not concern therevision of non-deterministic locating. Currently, there is no systematic study on how to deal with a fixture design thatfailed to provide deterministic location.2. Locating completeness evaluationIf deterministic location is not achieved by designed fixturing system, it is as important for designers to knowwhat the constraint status is and how to improve the design. If the fixturing 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 efficiently. 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;ci? and locating position xi;yi;zi? foreach locator, i 1;2;.;n; the n ? 6 locating matrix can be determined as follows:WLa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775.(7)When rankWL 6 and n 6; the workpiece is well-constrained.When rankWL 6 and n46; the workpiece is over-constrained. This means there are n ? 6 unnecessary locatorsin the locating scheme. The workpiece will be well-constrained without the presence of those n ? 6 locators. Themathematical representation for this status is that there are n ? 6 row vectors in locating matrix that can be expressedas linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist oneARTICLE IN PRESSTable 1RankNumber of locatorsStatuso 6Under-constrained 6 6Well-constrained 646Over-constrainedH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370locating scheme that provides deterministic location. The developed algorithm uses the following approach todetermine the unnecessary locators:1. Find all the combination of n ? 6 locators.2. For each combination, remove that n ? 6 locators from locating scheme.3. Recalculate the rank of locating matrix for the left six locators.4. If the rank remains unchanged, the removed n ? 6 locators are responsible for over-constrained status.This method may yield multi-solutions and require designer to determine which set of unnecessary locators shouldbe removed for the best locating performance.When rankWLo6; the workpiece is under-constrained.3. Algorithm development and implementationThe algorithm to be developed here will dedicate to provide information on un-constrained motions of theworkpiece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ARTICLE IN PRESSFig. 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 simplified locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378371orientation errors and locator errors can be expressed as follows:DX DxDyDzaxayaz2666666666437777777775w11:w1i:w1nw21:w2i:w2nw31:w3i:w3nw41:w4i:w4nw51:w5i:w5nw61:w6i:w6n2666666666437777777775?Dr1:Dri:Drn2666666437777775,(8)where Dx;Dy;Dz;ax;ay;azare displacement along x, y, z axis and rotation about x, y, z axis, respectively. Driisgeometric error of the ith locator. wijis defined by right generalized inverse of the locating matrix Wr WTLWLWTL?15.To identify all the un-constrained motions of the workpiece, V dxi;dyi;dzi;daxi;dayi;dazi? is introduced such thatV DX 0.(9)Since rankDXo6; there must exist non-zero V that satisfies 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;0? says that therotation about x-axis is not constrained. 0;1;1;0;0;0? means that the workpiece can move along the direction given byvector 0;1;1?: There could be infinite solutions. The solution space, however, can be constructed by 6 ? rankWLbasic solutions. Following analysis is dedicated to find out the basic solutions.From Eqs. (8) and (9)VX dxDx dyDy dzDz daxDax dayDay dazDaz dxXni1w1iDri dyXni1w2iDri dzXni1w3iDri daxXni1w4iDri dayXni1w5iDri dazXni1w6iDriXni1Vw1i;w2i;w3i;w4i;w5i;w6i?TDri 0.10Eq. (10) holds for 8Driif and only if Eq. (11) is true for 8i1pipn:Vw1i;w2i;w3i;w4i;w5i;w6i?T 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; w1j ?w2jfor 8j: In this scenario, the workpiece cannot move along x- or y-axis. However, itcan move along the diagonal line between x- and y-axis defined by vector 1, 1, 0.To find 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 ? r dimension space U u1:uj:un?rhi1pjpn ? r; 1pujpn: Select ujin the way that rankWL r still holds after setting all the terms of all the ujth row(s) equal to zero. Set r ? 6 modified locating matrixWLMa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775r?6,where i 1;2;:;niauj:ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 3683783722. Compute the 6 ? n right generalized inverse of the modified locating matrixWr WTLMWLMWTLM?1w11:w1i:w1rw21:w2i:w2rw31:w3i:w3rw41:w4i:w4rw51:w5i:w5rw61:w6i:w6r26666666664377777777756?r3. Trim Wrdown to a r ? rfull rank matrix Wrm: r rankWLo6: Construct a 6 ? r dimension vector Q q1:qj:q6?rhi1pjp6 ? r; 1pqjpn: Select qjin the way that rankWr r still holds after setting all theterms of all the qjth row(s) equal to zero. Set r ? r modified inverse matrixWrmw11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r26666664377777756?6,where l 1;2;:;6 laqj:4. Normalize the free motion space. Suppose V V1;V2;V3;V4;V5;V6? is one of the basic solutions of Eq. (10) withall six terms undetermined. Select a term qkfrom vector Q1pkp6 ? r: SetVqk ?1;Vqj 0 j 1;2;:;6 ? r;jak;(5. Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found asfollows.v1:vs:v62666666437777775wqk1:wqki:wqkr2666666437777775?w11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r2666666437777775?1,where s 1;2;:;6saqj;saqk;l 1;2;:;6 laqj:6. Repeat step 4 (select another term from Q) and step 5 until all 6 ? r basic solutions have been determined.Based on this algorithm, a C+ program was developed to identify the under-constrained status and un-constrained motions.Example 1. In a surface grinding operation, a workpiece is located on a fixture system as shown in Fig. 4. The normalvector and position of each locator are as follows:L1:0, 0, 10, 1, 3, 00,L2:0, 0, 10, 3, 3, 00,L3:0, 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.WL0013?100013?300011?20010?203010?2012666666437777775.ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378373This locating system provides under-constrained positioning since rankWL 5o6: The program then calculatesthe right generalized inverse of the locating matrix.Wr000000:50:5?1?0:51:50:75?1:251:5000:250:25?0:5000:5?0:50000000:5?0:526666666643777777775.The first row is recognized as a dependent row because removal of this row does not affect rank of the matrix. Theother five rows are independent rows. A linear combination of the independent rows is found according therequirement in step 5 of the procedure for under-constrained status. The solution for this special case is obvious that allthe coefficients are zero. Hence, the un-constrained motion of workpiece can be determined as V ?1; 0; 0; 0; 0; 0?: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.Example 2. 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, ?877, ?5150,L2:0, 1, 00, 1060, ?875, ?3780,L3:0, 1, 00, 1010, ?959, ?6120,L4:0.9955, ?0.0349, 0.0880, 977, ?902, ?6240,L5:0.9955, ?0.0349, 0.0880, 977, ?866, ?6240,L6:0.088, 0.017, ?0.9960, 1034, ?864, ?3590.The locating matrix of this configuration isWL010515:000:8960010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:9960866:6257998:24660:093626666666643777777775,rankWL 5o6 reveals that the workpiece is under-constrained. It is found that one of the first five rows can beremoved without varying the rank of locating matrix. Suppose the first row, i.e., locator L1is removed from WL; theARTICLE IN PRESSXZYL3L4L5L2L1Fig. 4. Under-constrained locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374modified locating matrix turns intoWLM010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:996866:6257998:24660:09362666666437777775.The right generalized inverse of the modified locating matrix isWr1:8768?1:8607?20:666521:37160:49953:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775.The program checked the dependent row and found every row is dependent on other five rows. Without losinggenerality, the first row is regarded as dependent row. The 5 ? 5 modified inverse matrix isWrm3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:028402666666437777775.The undetermined solution is V ?1; v2; v3; v4; v5; v6?:To calculate the five undetermined terms of V according to step 5,1:8768?1:8607?20:666521:37160:499526666666643777777775T?3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775?1 0; ?1:713; ?0:0432; ?0:0706; 0:04?.Substituting this result into the undetermined solution yields V ?1;0; ?1:713; ?0:0432; ?0:0706; 0:04?ARTICLE IN PRESSFig. 5. Knuckle 610 (modified from real design).H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378375This vector represents a free motion defined by the combination of a displacement along ?1, 0, ?1.713 directioncombined and a rotation about ?0.0432, ?0.0706, 0.04. To revise this locating configuration, another locator shouldbe added to constrain this free motion of the workpiece, assuming locator L1was removed in step 1. The program canalso calculate the free motions of the workpiece if a locator other than L1was removed in step 1. This provides morerevision options for designer.4. SummaryDeterministic location is an important requirement for fixture locating scheme design. Analytical criterion fordeterministic status has been well established. To further study non-deterministic status, an algorithm for checking thegeometry constraint status has been developed. This algorithm can identify an under-constrained status and indicatethe un-constrained motions of workpiece. It can also recognize an over-constrained status and unnecessary locators.The output information can assist designer to analyze and improve an existing locating scheme.Appendix. Locating matrixConsider a general workpiece as shown in Fig. 6. Choose reference frame fWg fixed to the workpiece. Let fGg andfLig be the global frame and the ith locator frame fixed relative to it. We haveFiXw;Hw;rwi fiXli;Hli;rli,(12)where Xw2 3?1and Hw2 3?1(Xli2 3?1and Hli2 3?1) are the position and orientation of the workpiece(the ith locator) in the global frame fGg; rwi2 3?1(rli2 3?1) is the position of the ith contact point between theworkpiece and the ith locator in the workpiece frame fWg (the ith locator frame fLig).Assume that DXw2 3?1(DHw2 3?1) and Drwi2 3?1are the deviations of the position Xw2 3?1(orientationHw2 3?1) of the workpiece and the position of the ith contact point rwi2 3?1; respectively. Then we have the actualcontact on the workpiece,FiXw DXw;Hw DHw;rwi Drwi FiXw;Hw;rwi qFiqXwDXwqFiqHwDHwqFiqrwiDrwi,13where the second term in the right side of Eq. (13) is the position error of the ith contact point resulting from theposition error DXwof the workpiece, the third term is the position error of the ith contact point resulting from theorientation error DHwof the workpiece, and the fourth term is the position error of the ith contact point resulting fromits workpiece geometric variation Drwion the workpiece.Similarly, assume that DXli2 3?1DHli2 3?1 and Drli2 3?1are the deviations of the position Xli2 3?1(orientation Hli2 3?1) of the ith locator and the position of the ith contact point rli2 3?1; respectively. The actualcontact on the ith locator isfiXli DXli;Hli DHli;rli Drli fiXli;Hli;rli qfiqXliDXliqfiqHliDHliqfiqrliDrli,14ARTICLE IN PRESSworkpieceith locator Win iLiwrilr GilXwXwX wFig. 6. Fixture coordinate frames.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378376where the second term in the right side of Eq. (14) is the position error of the ith contact point resulting from theposition error DXliof the ith locator, the third term is the position error of the ith contact point resulting from theorientation error DHliof the
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