鉆套零件的工藝夾具設計

資源目錄里展示的全都有，所見即所得。下載后全都有,請放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 河南工業(yè)職業(yè)技術學院 機床夾具設計 課程設計說明書 設計題目：設計“鉆套夾具設計”鉆孔 班 級 設 計 者 指導教師 設計日期 目錄 序言 1 1夾具設計任務書 2 1.1零件 2 1.2技術要求分析 2 2確定夾具設計方案 2 2.1 基準面選擇 2 2.2 定位方式 3 2.3夾緊方案、夾緊力計算 3 2.4分度方案 4 3夾具設計 4 3.1夾具圖上的尺寸、公差和技術要求 4 3.1.1夾具圖上技術要求標注 4 3.1.2夾具圖上公差值的確定 4 3.2對刀方案確定 5 4誤差分析與計算 5 5主要零件的設計說明 5 5.1夾具體 5 5.2其他元件 6 6夾具裝配要求 7 7夾具的使用維護要求 7 7.1夾具的使用 7 7.2夾具的使用維護要求 7 總結 8 參考文獻 9 夾具圖 10 夾具裝配圖 11 課程設計心得體會 12 序???言 機械制造業(yè)是制造具有一定形狀位置和尺寸的零件和產品，并把它們裝備成機械裝備的行業(yè)。機械制造業(yè)的產品既可以直接供人們使用，也可以為其它行業(yè)的生產提供裝備，社會上有著各種各樣的機械或機械制造業(yè)的產品。我們的生活離不開制造業(yè)，因此制造業(yè)是國民經濟發(fā)展的重要行業(yè)，是一個國家或地區(qū)發(fā)展的重要基礎及有力支柱。從某中意義上講，機械制造水平的高低是衡量一個國家國民經濟綜合實力和科學技術水平的重要指標。 鉆模的加工工藝規(guī)程及其，鉆孔的夾具設計是在學完了機械制圖、機械制造技術基礎、機械設計、機械工程材料等進行課程設計之后的下一個教學環(huán)節(jié)。正確地解決一個零件在加工中的定位，夾緊以及工藝路線安排，工藝尺寸確定等問題，并設計出專用夾具，保證零件的加工質量。本次設計也要培養(yǎng)自己的自學與創(chuàng)新能力。因此本次設計綜合性和實踐性強、涉及知識面廣。所以在設計中既要注意基本概念、基本理論，又要注意生產實踐的需要，只有將各種理論與生產實踐相結合，才能很好的完成本次設計。本次設計水平有限，其中難免有缺點錯誤，敬請老師們批評指正。 12 1 夾具設計任務書 1.1 零件圖 1.2 技術要求分析 零件的技術要求主要包括尺寸精度、形狀精度、位置精度、表面粗糙度要求等，這些技術要求應當是能夠保證零件使用性能前提下的極限值。進行零件技術要求分析，主要是分析這些技術要求的合理性，以及實現(xiàn)的可能性，重點分析重要表面和部位的加工精度和技術要求，為制定合理的加工方案做好準備。同時通過分析以確定技術要求是否過于嚴格，因為過高的精度和過小的表面粗糙度要求會使工藝過程變得復雜，加工難度大，增加不必要的成本。 從圖得知，該零件的部分尺寸精度要求較高，最高公差為0.06mm，在加工中可以達到此要求；該零件無形位公差要求，按照IT10—11進行控制均可；零件的表面粗糙度要求不高，最高處才Ra3.2，在加工中很容易保證。 2 確定夾具設計方案 2.1 基準面的選擇 由于零件有一個較大的中心孔，所以可以采用定位元件進行定位,而又零件的下表面是設計基準,且已經加工完,所以決定采用短銷和一平面進行定位,短銷限制2個自由度分別為X方向和Y方向的自由度,而一平面限制3個自由度，分別為繞X轉,繞Y轉和Z方向，而由于零件比較復雜所以用三個支撐釘來充當一個平面，限制3個自由度，零件為Z方向加工孔，但不是中心孔，需要限制5個自由度，所以還需選擇一個繞Z方向旋轉的定位，而可以同過一個一個活動型塊來通過來限制此自由度，選用的具體定位裝置見裝配圖。 加工一開始，總是把用作精加工基準的表面加工出來，因為定位基準的表面精確，裝夾誤差就小，所以任何零件的加工過程，總是先對定位基準面進行粗加工和半精加工，必要時還要進行精加工，例如，軸類零件總是對定位基準面進行粗加工和半精加工，再進行精加工。例如軸類零件總是先加工中心孔，再以中心孔面和定位孔為精基準加工孔系和其他表面。如果精基準面不止一個，則應該按照基準轉換的順序和逐步提高加工精度的原則來安排基準面的加工。 2.2 定位方式 根據零件鉆孔的尺寸、形狀和位置嫉精度要求，工件定位時需限制5個自由度。工件的定位基準和夾緊位置雖然在工序圖上已經規(guī)定，但在擬定夾緊方案時，扔需要對其進行分析研究，主要是考察定位基準的選擇能否滿足工件位置精度的要求，夾具的結構能否實現(xiàn)。 在鉆削的工序中，工件在鉆孔方向的工序基準是和孔平行的面，若以此端面為定位基準，可以達到與工序基準想重合。但是，由于要在次面上鉆孔，那么夾具的定位面就要設計成另外一個面，這樣會給定位和夾緊帶來麻煩，夾具架構也比較復雜。如果選擇與所加工孔的相對另一端面為定位基準，則會引起基準不重合誤差，其大小等于工件面端面之間的公差，考慮到孔的公差稍微大些，因此完全可以保證加工精度的要求。同時，這樣可以使定位夾緊可靠，操作方便，所以應選擇工件底面為定位基準，采用定位銷作為定位元件。 2.3 夾緊方案、夾緊力計算 根據夾緊力的方向應朝向主要限位面以及作用點應落在應落在定位元件的支撐范圍內的原則，夾緊力的作用線應落在中間范圍內，夾緊力與垂直的夾角應盡量小，以保證夾緊穩(wěn)定可靠。 由于鉆套不是很長，所以用一塊壓板夾緊。如果采用手動夾緊，工件裝卸所花的時間較長，不能適應大批生產的要求。夾緊元件主要是定位銷，由于該工件較小，批量又不大，為使夾具結構簡單，采用了手動的螺旋壓板夾緊機構。 螺紋公稱直徑d =10mm 螺紋中徑d2=9mm 手柄長度L=120mm 作用力FQ=50N 計算的FW=9379N 2.4分度方案 由于4個通孔的對稱度要求不高(未標注公差)，設計一般精度的分度裝置即可。分度盤與定位心軸做成一體，在夾具體的回轉套中回轉，采用對定銷對定，鎖緊螺母鎖緊，結構簡單，動作迅速可靠。 3 夾具設計 3.1夾具圖上的尺寸、公差和技術要求 3.1.1夾具圖上技術要求標注 ?為了保證夾具制造和裝配后達到設計規(guī)定的精度要求，在設計圖上除了直接標注尺寸公差和形位公差外，夾具總圖上無法用符合標準而又必須說明的問題，可作為技術要求用文字寫在總圖上，習慣上把用文字說明的夾具精度要求統(tǒng)稱為技術條件。主要內容有：? 1）夾具的裝配、調整方法。? 2）某些零件的重要表面應一起加工。3）夾具表面上的裝飾要求。 4）夾具使用時的操作順序。 3.1.2夾具圖上公差值的確定 夾具總圖上標注公差值的原則： 在滿足工件加工要求的前提下，盡量降低夾具的制造精度。直接影響工件加工精度的夾具公差δJ。取夾具總圖上的尺寸公差或δJ?=(1/2~1/5)?，式中δk一與δJ相應的工件尺寸公差或位置公差。當工件批量大、精度低時，δJ取小值，反之取最大。? 對于直接影響工件加工精度的配合尺寸，在確定了配合性質后，應盡量選用優(yōu)先配合。 ?工件的加工尺寸未注公差時，工件公差δk視為IT12~IT14，夾具上相應的尺寸公差按IT9~IT11標注；工件上的位置要求未注公差10，工件位置公差δk視為9~11級，夾具上的相應位置公差按7~9級標注；工件上加工角度未注公差時，工件公差δk視為±30~±10，夾具上相應的角度公差標為10~±3（相應邊長為10~400mm，邊長短時取最大值）。??夾具上其他重要尺寸的公差與配合。以上各尺寸和公差、技術要求、公差值都在夾具體中和夾具裝備總圖中進行了具體的標注，詳見附錄中的夾具體圖樣和夾具裝配總圖。 3.2對刀方案的確定 本工序被加工的孔精度一般，主要保證孔和空的中心線通過大孔 Φ22h7中心等要求。夾具中采用對刀塊及塞尺的對刀裝置來調整鉆刀相對夾具的位置。其中，利用對刀塊的鉛垂對刀面級塞尺調整鉆刀，使其寬度方向的對稱面通過圓柱銷的中心，從而保證該零件加工后能達到滿足要求。加工孔的鉆刀需要兩個方向對刀，故采用直角對刀塊。 4 誤差分析與計算 △D=△B+△Y,式中△D-定位誤差； △B-基準不重合誤差； △Y-基準位移誤差。 由于軸套孔的工序基準是15和定位基準是軸線，基準不重合，△B=0.025；夾具中定位基準面是Φ22軸線，所以存在基準位移誤差：△Y=0.013mm 定位誤差：△B=0.025mm △D=△Y+△B=0.038mm 工件尺寸25mm公差按自由公差，取ITl2級，查表IT=0.21mm，所以加工公差=0.21mm,△D<／3,此定位方案可行。 因為我們我們要加工的面沒有位置度要求，同時也沒有公差要求，因此一般的定位都能滿足要求，需要考慮如何提高生產效率。 5 主要零件的設計說明 5.1 夾具體 選用鑄造夾具體，夾具體是夾具中的基礎元件，它所有的元件的支撐面，分度盤的鎖緊裝置在夾具體的下方，因此夾具體下方要挖出槽來。 夾緊力分析：夾緊力是保證定位穩(wěn)定,夾具可行的因素.夾緊力不能太小，否則加工時容易發(fā)生工件位移而破壞定位。夾緊力也不太過大，否則工件易變形，增大結構尺寸。本次鉆孔孔徑較小，所需夾緊力較??；綜合考慮加工因素，選用M10的夾緊螺母，能夠滿足夾緊要求。 5.2 其他元件 6 夾具裝配要求 (1)總裝配圖應按國家標準盡可能1:1的繪制，這樣圖樣有良好的直觀性。主視圖應按操作實際位置布置，三制圖要能完整清出表示出夾具的工作原理和結構。 (2)視工件為透明體，用雙點畫線畫出主要部件。畫出定位元件、夾緊機構、導向裝置的位置。 （3）按照夾緊狀態(tài)畫出夾緊元件和夾緊機構。 （4）標注主要的尺寸、配合、公差等。 7 夾具的使用維護要求 7.1夾具的使用要求 （1）使用前對限位尺寸檢查是否還保持正確位置； （2）如果擋銷磨損超差，可以進行打磨修復；如果擋板、插銷、定位錐頭銷磨損超差，可以重新組裝，錯開磨損部位后繼續(xù)使用； （3）使用后需要涂防銹油 （4）夾具應清潔定位面，不許積聚污垢和其他銹蝕物，夾緊時要求平穩(wěn)，不使工件移位 7.2夾具的維護要求 常用的夾具，要保護定位面不被劃傷，兩件夾具可定期放到平面磨床上磨平修整，使二者保持等高。對于夾具絕緣性的測定也應注意，尤其是絕緣夾層用吸水性較大的材料制成的夾具. 總 結 一周的課程設計結束了，時間雖然短暫但是它對我們來說受益菲淺的，通過這次的設計使我們不再是只知道書本上的空理論，不再是紙上談兵，而是將理論和實踐相結合進行實實在在的設計，使我們不但鞏固了理論知識而且掌握了設計的步驟和要領，使我們更好的利用圖書館的資料，更好的更熟練的利用我們手中的各種設計手冊和AUTOCAD等制圖軟件，為我們踏入設計打下了好的基礎。 在這次的課程設計中不僅檢驗了我所學習的知識，也培養(yǎng)了我如何去把握一件事情，如何去做一件事情，又如何完成一件事情。在設計過程中，與同學商量分析，和同學相互探討，相互學習，相互監(jiān)督。學會了合作，學會了寬容，學會了理解。 課程設計是我們專業(yè)課程知識綜合應用的實踐訓練，這是我們邁向社會，從事職業(yè)工作前一個必不少的過程。我今天認真的進行課程設計，學會腳踏實地邁開這一步，就是為明天穩(wěn)健地在社會大潮中奔跑打下堅實的基礎。通過這次夾具設計，本人在多方面有所提高。通過這次夾具設計，綜合運用本專業(yè)所學課程的理論和提高學生獨立工作能力，鞏固與擴充了鉆模夾具設計等課程所學內容，掌握了鉆模夾具設計的方法和步驟，掌握了鉆模夾具設計的基本的夾具技能懂得了怎樣分析零件的工藝性，怎樣確定工藝方案，了解了夾具的基本結構，提高了計算能力，繪圖能力，熟悉了規(guī)范和標準，同時各科相關的課程都有了全面的復習，獨立思考的能力也有了提高。 課程設計使我們認識到了努力的學好書本上的知識是不夠的，還應該更好的做到理論和實踐的結合。因此同學們非常感謝老師給我們的辛勤指導，使我們學到了好多，也非常珍惜學院給我們的這次設計的機會，它將是我們將來走向工作崗位的更出色的關鍵一步。 參考資料 1.切削用量簡明手冊,艾興、肖詩綱主編,機械工業(yè)出版社出版，1994年 2.機械制造工藝設計簡明手冊，李益民主編，機械工業(yè)出版社出版，1994年 3.機床夾具設計，機械工業(yè)大學、上海工業(yè)大學主編、上?；瘜W工業(yè)出版社、上?？茖W技術出版社出版，1983年 4.機床夾具設計手冊，東北重型機械學院、洛陽工學院、一汽制造廠職工大學編，上?？茖W技術出版社出版，1990年 5.金屬機械加工工藝人員手冊，上?？茖W技術出版社，1981年10月 6.機械制造工藝學，郭宗連、秦寶榮主編，中國建材工業(yè)出版社出版，1997年 7.趙家奇,機械制造工藝學課程設計指導書(2版),機械工業(yè)出版社,2006年.? 8.曾志新,呂明主編,機械制造技術基礎,:武漢理工大學出版社,2001年.? 9.李益明主編,機械制造工藝設計簡明手冊機械工業(yè)出版社,1993年.? 9.肖詩綱主編,切削用量手冊,機械工業(yè)出版社,1993年.? 10.金屬切削機床夾具設計手冊.上海柴油機廠工藝設備研究所編,機械工業(yè)出版社， 1987年 3課程設計心得體會 我所選課程設計題題目是“鉆模夾具鉆孔”，對于我自己來說，它是很具有的挑戰(zhàn)性的，總覺得自己所學到的知識很欠缺，但是越是自己薄弱的環(huán)節(jié)越要有勇氣去嘗試。在課程設計的工藝設計的過程中，因為沒有真正的工作經驗，在做夾具設計的工程中總覺得很辛苦，有時還會產生放棄的念頭，但是最終堅持了下來，完成了我的課程設計，為了自己的目標，更為了自己的選擇。在指導老師的指點下，我開始通過各種渠道翻閱資料。接下來，我開始對所搜集的資料進行整理、分析研究，解決設計中的問題，最終在短暫的一周時間內順利完成任務。 通過這次的課程設計，使我深深體會到，干任何事情都必須認真、耐心、細致。課程設計過程中，許多計算有時候令我感到有些心煩意亂，有幾次因為疏忽，數據算錯，只能毫不留情地重新來。但一想起老師平時對我們耐心的教導，想到不久的將來自己就要承擔社會責任，想到世界上有好多因為很小的細節(jié)失誤而出現(xiàn)的令人無比震驚的事故，無時不在提醒著自己，一定要養(yǎng)成一種高度負責、一絲不茍的良好習慣。 至此，我衷心的感謝老師。是你的嚴厲批評喚醒了我，是您的諄諄教誨啟發(fā)了我，我感謝老師您今天又為我增添了一副堅強的翅膀。 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