汽車(chē)前拖鉤鉆削φ17mm、φ13mm、Ra3.2粗糙度兩孔夾具設(shè)計(jì)【含9張CAD圖紙+PDF圖】,含9張CAD圖紙+PDF圖,汽車(chē),前拖鉤鉆削,17,mm,13,Ra3,粗糙,度兩孔,夾具,設(shè)計(jì),CAD,圖紙,PDF 畢 業(yè) 論 文 中 文 摘 要 摘要：夾具是機(jī)械加工不可缺少的部件，在機(jī)床技術(shù)向高速、高效、精密、復(fù)合、智能、環(huán)保方向發(fā)展的帶動(dòng)下，夾具技術(shù)正朝著高精、高效、模塊、組合、通用、經(jīng)濟(jì)方向發(fā)展。 該畢業(yè)設(shè)計(jì)任務(wù)是鉆削汽車(chē)前拖鉤φ17mm、φ13mm、Ra3.2粗糙度的兩個(gè)通孔。工件以臺(tái)階面、側(cè)面和圓弧面為定位基準(zhǔn)，工件材料45＃鋼，數(shù)量為中批量。由于該零件屬于中批量生產(chǎn)，鉆孔精度有要求，因此設(shè)計(jì)一個(gè)專(zhuān)用的鉆床夾具，保證零件的加工質(zhì)量，提高加工效率。 本夾具采用移動(dòng)式結(jié)構(gòu)是為了工件在一次安裝中鉆兩較大的孔時(shí)，可以極為方便地確定鉆套對(duì)刀具的位置，從而節(jié)省找正夾具的時(shí)間，減少鉆套與刀具的磨損。 完成鉆床夾具的所有設(shè)計(jì)后，用PRO/E進(jìn)行三維圖形的繪制，首先畫(huà)裝配圖，然后從裝配圖上拆畫(huà)零件圖，標(biāo)注相關(guān)尺寸及技術(shù)要求，最后進(jìn)行論文的撰寫(xiě)、整理、修改完成該畢業(yè)設(shè)計(jì)。 關(guān)鍵詞 夾具；鉆床夾具；精度；CAD 畢 業(yè) 論 文 外 文 摘 要 Title Car ex- design for dragging along hook to drill a hole tongs Abstract The tongs is a machine to process an indispensable parts, at tool machine technique to the high speed, efficiently, nicety, compound, intelligence, environmental protection direction development of arouse under, tongs technique just toward the high Jing, efficiently, mold piece, combination, in general use and economic direction development. The graduation design's task is two bores that drill to drag along to hook φ 17 mms, φ 13 mms and Ra3.2 roughnesses before paring a car.The work piece takes step noodles, on the side and arc noodles as to position basis, the work piece material is 45# steels, amount is medium batch quantity.That spare parts belongs to medium batch quantity production and drills a hole accuracy to contain request, therefore design drilling of an appropriation bed tongs, promise processing of spare parts quality, the exaltation processes an efficiency. This tongs adoption ambulation type structure for the sake of work piece while drilling two bigger bores in a gearing, can extremely and expediently make sure to drill a set to the position that the knife has, save time of seeking the positive tongs thus, the decrease drills set and knife to have of wear away. The completion after drilling all designs of bed tongs carries on drawing of two-dimensional sketch with AutoCAD, the painting assembles diagram first, then from assemble the diagram up dismantle painting spare parts diagram and mark related size and technique request, finally carrying on a compose, tidying up of thesis and modifying completion should graduate a design. Keywords Fixture; drilling fixture; accuracy; CAD 學(xué)生畢業(yè)設(shè)計(jì)（論文）中期檢查表 學(xué)生姓名 學(xué)號(hào) 指導(dǎo)教師 選題情況 課題名稱(chēng) 汽車(chē)前拖鉤鉆孔夾具設(shè)計(jì) 難易程度 偏難 適中 偏易 工作量 較大 合理 較小 符合規(guī)范化的要求 任務(wù)書(shū) 有 無(wú) 開(kāi)題報(bào)告 有 無(wú) 外文翻譯質(zhì)量 優(yōu) 良 中 差 學(xué)習(xí)態(tài)度、出勤情況 好 一般 差 工作進(jìn)度 快 按計(jì)劃進(jìn)行 慢 中期工作匯報(bào)及解答問(wèn)題情況 優(yōu) 良 中 差 中期成績(jī)?cè)u(píng)定： 指導(dǎo)教師意見(jiàn)： 指導(dǎo)教師： 年 月 日 畢業(yè)設(shè)計(jì)(論文)任務(wù)書(shū) 教 學(xué) 點(diǎn)： 湖南科技工業(yè)職業(yè)技術(shù)學(xué)院 專(zhuān) 業(yè)： 機(jī)械電子工程 學(xué) 生 姓 名： 學(xué) 號(hào)： 設(shè)計(jì)（論文）題目： 汽車(chē)前拖鉤鉆孔夾具設(shè)計(jì) 起 訖 日 期： 2014年1月12日~ 4月15日 設(shè)計(jì)（論文）地點(diǎn)： 指 導(dǎo) 教 師： 專(zhuān) 業(yè) 負(fù) 責(zé) 人： 發(fā)任務(wù)書(shū)日期： 2014 年 1 月 12日 畢業(yè)設(shè)計(jì)（論文）任務(wù)書(shū) 1．本畢業(yè)設(shè)計(jì)（論文）課題應(yīng)達(dá)到的目的： （1）培養(yǎng)學(xué)生綜合運(yùn)用所學(xué)知識(shí)，結(jié)合實(shí)際獨(dú)立完成課題的工作能力。 （2）是對(duì)學(xué)生的知識(shí)面、掌握知識(shí)的深度、運(yùn)用理論結(jié)合實(shí)際去處理問(wèn)題的能力、實(shí)驗(yàn)?zāi)芰?、?jì)算機(jī)運(yùn)用水平、書(shū)面及口頭表達(dá)能力進(jìn)行的考核。 （3）畢業(yè)設(shè)計(jì)要求學(xué)生學(xué)會(huì)查閱、使用各種專(zhuān)業(yè)資料、軟件仿真、網(wǎng)上資源，并以嚴(yán)肅認(rèn)真、深入研究的工作作風(fēng)完成設(shè)計(jì)任務(wù)，促使學(xué)生向工程技術(shù)人員轉(zhuǎn)變。 2．本畢業(yè)設(shè)計(jì)（論文）課題任務(wù)的內(nèi)容和要求（包括原始數(shù)據(jù)、技術(shù)要求、工作要求等）： 如下圖，工件以臺(tái)階面、側(cè)面和圓弧面為定位基準(zhǔn)，需鉆削φ17mm、φ13mm、Ra3.2粗糙度的兩孔，工件材料45＃鋼，數(shù)量為為中批量。 畢業(yè)設(shè)計(jì)（論文）任務(wù)書(shū) 3．本畢業(yè)設(shè)計(jì)（論文）課題成果的要求：【包括畢業(yè)設(shè)計(jì)論文、圖表、實(shí)物樣品等】 1、繪制夾具總裝圖一張（A1圖）、全套零件圖（A4圖）； 2、確定和標(biāo)注夾具的技術(shù)要求及其允差值。 3、設(shè)計(jì)說(shuō)明書(shū)1份（包括零件圖分析、定位方案確定、定位誤差計(jì)算等內(nèi)容）6000字以上。 4．推薦參考資料： [1] 孟憲棟.機(jī)床夾具圖冊(cè).北京：機(jī)械工業(yè)出版社，2011.10； [2] 李名望.機(jī)床夾具設(shè)計(jì)實(shí)例教程.北京：化學(xué)工業(yè)出版社，2009.8； [3] 譚雪松. Pro/ENGINEER中文野火版4.0項(xiàng)目教程.北京：人民郵電出版社，2009.5； [4] 傅成昌,傅曉燕.公差與配合問(wèn)答.北京：機(jī)械工業(yè)出版社，2007.1； [5] 虞自?shī)^.AutoCAD2008高級(jí)案例解析.北京：中國(guó)電力出版社，2008.5； [6] 曹巖.機(jī)床夾具手冊(cè)與三維圖庫(kù).北京：化學(xué)工業(yè)出版社，2010.3； 畢業(yè)設(shè)計(jì)（論文）任務(wù)書(shū) 5．本畢業(yè)設(shè)計(jì)（論文）課題工作計(jì)劃進(jìn)度： 起訖日期 工作內(nèi)容 2014年 1月12日 ～ 2月18日 2月19日 ～ 2月 25日2月26日 ～ 3月15日 3月 16 日 ～ 4月9日 4月10日 ～ 4月15日 待定 查閱文獻(xiàn) 填寫(xiě)開(kāi)題報(bào)告 銑床夾具設(shè)計(jì)及計(jì)算，繪制裝配圖及零件圖 寫(xiě)設(shè)計(jì)論文，整理、完善、裝訂成冊(cè) 上交論文 論文答辯 所在專(zhuān)業(yè)審查意見(jiàn)： 負(fù)責(zé)人： 年 月 日 南 京 理 工 大 學(xué) 畢業(yè)設(shè)計(jì)（論文）前期工作材料 學(xué)生姓名： 學(xué) 號(hào)： 教 學(xué) 點(diǎn)： 湖南科技工業(yè)職業(yè)技術(shù)學(xué)院 專(zhuān) 業(yè)： 機(jī)械電子工程 設(shè)計(jì)題目： 汽車(chē)前拖鉤鉆孔夾具設(shè)計(jì) 指導(dǎo)教師： 副教授 2014年 3 月 參考文獻(xiàn) [1] 孟憲棟.機(jī)床夾具圖冊(cè).北京：機(jī)械工業(yè)出版社，2011.10； [2] 李名望.機(jī)床夾具設(shè)計(jì)實(shí)例教程.北京：化學(xué)工業(yè)出版社，2009.8； [3] 譚雪松. Pro/ENGINEER中文野火版4.0項(xiàng)目教程.北京：人民郵電出版社，2009.5； [4] 傅成昌,傅曉燕.公差與配合問(wèn)答.北京：機(jī)械工業(yè)出版社，2007.1； [5] 虞自?shī)^.AutoCAD2008高級(jí)案例解析.北京：中國(guó)電力出版社，2008.5； [6] 曹巖.機(jī)床夾具手冊(cè)與三維圖庫(kù).北京：化學(xué)工業(yè)出版社，2010.3； 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 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: hsongwpi.edu (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 wor