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河南理工大學(xué)本科畢業(yè)設(shè)計(jì)（論文）中期檢查表 指導(dǎo)教師： 姜無(wú)疾 職稱： 講師 所在系部（單位）： 機(jī)械與動(dòng)力工程學(xué)院 教研室（研究室）： 機(jī) 制 題目名稱 電機(jī)座液壓夾緊粗鏜夾具 姓名 李太平 專業(yè)班級(jí) 機(jī)制02－3班 學(xué)號(hào) 02080091 一、選題質(zhì)量 （主要從以下四個(gè)方面填寫：1、選題是否符合專業(yè)培養(yǎng)目標(biāo)，能否體現(xiàn)綜合訓(xùn)練要求； 2、題目難易程度；3、題目工作量；4、題目與生產(chǎn)、科研、經(jīng)濟(jì)、社會(huì)、文化及實(shí)驗(yàn)室建 設(shè)等實(shí)際的結(jié)合程度） 所選做的電機(jī)座液壓夾緊粗鏜夾具設(shè)計(jì)基本符合本專業(yè)的的培養(yǎng)目標(biāo)，基本能達(dá)到綜合訓(xùn)練的要求。設(shè)計(jì)的難度中等，圖紙復(fù)雜，計(jì)算量一般，工作量完全達(dá)到了畢業(yè)設(shè)計(jì)的要求。電機(jī)座液壓夾緊粗鏜夾具是電機(jī)座加工的重要輔助設(shè)備，同現(xiàn)實(shí)生活結(jié)合非常的密切，對(duì)我們?cè)O(shè)計(jì)來(lái)說(shuō)是個(gè)不錯(cuò)的題目。 二、開(kāi)題報(bào)告完成情況 已經(jīng)有了一定設(shè)計(jì)成果，確定了總體設(shè)計(jì)任務(wù)，設(shè)計(jì)方案也已經(jīng)確定?？傮w設(shè)計(jì)正在 設(shè)計(jì)當(dāng)中。整理出概述和基本目錄情況，完成了整體裝配圖和幾張重要零件圖的CAD繪制。 三、階段性成果 已初步對(duì)夾具的發(fā)展，以及目前國(guó)內(nèi)外的夾具的發(fā)展有了一定的了解，對(duì)夾具的重要結(jié)構(gòu)和功能有了較為細(xì)致的認(rèn)識(shí)，對(duì)它的各個(gè)組成部分進(jìn)行了分析，完成了一部分零件圖和說(shuō)明書，對(duì)該課題也有了更進(jìn)一步的認(rèn)識(shí)；其工作原理和主要的部件已經(jīng)比較熟悉。 四、存在主要問(wèn)題 現(xiàn)在手頭的資料還是非常少，急切需要尋找更多的資料以完成后面的設(shè)計(jì)，另外對(duì)個(gè)別零件還有待做更深層次的認(rèn)識(shí)和把握，希望能夠得到老師的進(jìn)一步指導(dǎo)。 五、指導(dǎo)教師對(duì)學(xué)生在畢業(yè)實(shí)習(xí)中，勞動(dòng)、學(xué)習(xí)紀(jì)律及畢業(yè)設(shè)計(jì)（論文）進(jìn)展等方面的評(píng)語(yǔ) 該生在設(shè)計(jì)的過(guò)程中，能夠認(rèn)識(shí)到畢業(yè)設(shè)計(jì)的重要性，能積極準(zhǔn)備，嚴(yán)格要求。在設(shè)計(jì)時(shí)，能夠很好的和導(dǎo)師保持聯(lián)系，與同組同學(xué)通力合作。在整個(gè)設(shè)計(jì)過(guò)程中表現(xiàn)良好。 指導(dǎo)教師： （簽名） 年 月 日 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)（論文）開(kāi)題報(bào)告 題目名稱 電機(jī)座液壓夾緊粗鏜夾具 學(xué)生姓名 李太平 專業(yè)班級(jí) 機(jī)制02-3班 學(xué)號(hào) 02080091 一、選題的目的和意義 1 能夠掌握設(shè)計(jì)計(jì)算的基本原理和方法,提高設(shè)計(jì)計(jì)算的能力. 2 加深領(lǐng)會(huì)計(jì)算的基本理論和深化所學(xué)的理論知識(shí). 3 樹立正確的設(shè)計(jì)思想，為以后在工作中遇到相關(guān)問(wèn)題提供解決依據(jù). 通過(guò)本次畢業(yè)設(shè)計(jì)，能使我們把先修的基礎(chǔ)和專業(yè)基礎(chǔ)課程中所獲得的理論知識(shí)在實(shí)際的設(shè)計(jì)工作中綜合地加以應(yīng)用，通過(guò)畢業(yè)設(shè)計(jì)之后能夠熟練應(yīng)用有關(guān)參考資料、計(jì)算圖表、手冊(cè)；熟悉有關(guān)的國(guó)家標(biāo)準(zhǔn)和部頒標(biāo)準(zhǔn)，為以后成為優(yōu)秀的工程技術(shù)人員打下良好的基礎(chǔ) 二、國(guó)內(nèi)外研究綜述 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)正朝著精密化、自動(dòng)化、標(biāo)準(zhǔn)化、通用化、專業(yè)化方向發(fā)展。 　　精密化 當(dāng)前機(jī)械制造業(yè)精加工和超精加工迅速發(fā)展，高精度機(jī)床大量出現(xiàn)。隨之夾具的精度也必然提高，精密夾具也越來(lái)越多，并直接用于生產(chǎn)線上。 0.01mm/300mm，平行度高達(dá)0.01mm/500mm。德國(guó)demmeler(戴美樂(lè))公司制造的4m長(zhǎng)、2m寬的孔系列組合焊接夾具平臺(tái)，其等高誤差為±0.03mm；精密平口鉗的平行度和垂直度在5μm以內(nèi)；夾具重復(fù)安裝的定位精度高達(dá)±5μm；瑞士EROWA柔性?shī)A具的重復(fù)定位精度高達(dá)2~5μm。機(jī)床夾具的精度已提高到微米級(jí)，世界知名的夾具制造公司都是精密機(jī)械制造企業(yè)。 自動(dòng)化 在實(shí)現(xiàn)工藝過(guò)程自動(dòng)化時(shí)，夾具也必須實(shí)現(xiàn)自動(dòng)化。目前除了在生產(chǎn)流水線和自動(dòng)線上為通用機(jī)床和專用機(jī)床配置自動(dòng)化夾具外，在數(shù)控機(jī)床上，尤其在加工中心上出現(xiàn)了各種自動(dòng)化夾具。當(dāng)前，國(guó)外柔性制造系統(tǒng)（FMS）發(fā)展迅速，自動(dòng)化隨行夾具水平越來(lái)越高，不但出現(xiàn)了刀具庫(kù)也出現(xiàn)了夾具庫(kù)。 標(biāo)準(zhǔn)化 為了提高夾具在設(shè)計(jì)﹑制造和使用上的經(jīng)濟(jì)效益，我國(guó)夾具元件標(biāo)準(zhǔn)化工作已經(jīng)有較快的進(jìn)展。中型組合夾具的元件已有了國(guó)際標(biāo)準(zhǔn)。不少工廠也在生產(chǎn)標(biāo)準(zhǔn)化規(guī)格化的夾緊動(dòng)力裝置（如氣缸﹑油缸等）。 通用化 為了適應(yīng)多種品種小批量的生產(chǎn)特點(diǎn)，在標(biāo)準(zhǔn)化﹑規(guī)格化的基礎(chǔ)上大力發(fā)展了可調(diào)整通用夾具和成組夾具。夾具的功能可根據(jù)工件的幾何要素及工藝要素的相似性做到一套夾具多種用途。德國(guó)demmeler(戴美樂(lè))公司的孔系列組合焊接夾具，僅用品種、規(guī)格很少的配套元件，即能組裝成多種多樣的焊接夾具。元件的功能強(qiáng)，使得夾具的通用性好，元件少而精，配套的費(fèi)用低，經(jīng)濟(jì)實(shí)用才有推廣應(yīng)用的價(jià)值。 專業(yè)化 在生產(chǎn)技術(shù)準(zhǔn)備工作中，夾具的設(shè)計(jì)與制造占了很大比例。其準(zhǔn)備周期拖得很長(zhǎng)，不利于市場(chǎng)競(jìng)爭(zhēng)。為了縮短生產(chǎn)準(zhǔn)備周期，目前有些夾具得設(shè)計(jì)與制造任務(wù)由專業(yè)化工廠來(lái)完成。我國(guó)組合夾具已向全國(guó)大量供貨，并向國(guó)外出口槽系標(biāo)準(zhǔn)化元件。夾具的專業(yè)化生產(chǎn)不但能提高經(jīng)濟(jì)效益而且可加速夾具專業(yè)的技術(shù)現(xiàn)代化，是今后的發(fā)展方向。 加強(qiáng)機(jī)床夾具方向的基礎(chǔ)研究 機(jī)床夾具除上述各方面發(fā)展趨勢(shì)外，還有一些有關(guān)夾具的基礎(chǔ)理論和制造等方面的問(wèn)題值得研究，如工件的定位理論，夾具的精度（或誤差）分析及其加工總誤差的關(guān)系。夾具合理精度的制定，夾具元件磨損規(guī)律和磨損標(biāo)準(zhǔn)的制定等都是值得研究的基本問(wèn)題。 此外，夾具新結(jié)構(gòu)新元件÷新材料等方面的研究和發(fā)展也應(yīng)給予足夠的重視。 三、畢業(yè)設(shè)計(jì)所用的方法 1 在學(xué)校圖書館查閱相關(guān)資料。 2 在工廠的實(shí)踐畢業(yè)實(shí)習(xí)。 3 通過(guò)老師和工程師的指導(dǎo)。 4 通過(guò)瀏覽因特網(wǎng)上的相關(guān)資料。 5 通過(guò)對(duì)相關(guān)資料和數(shù)據(jù)的理論計(jì)算和分析 四、主要參考文獻(xiàn)與資料獲得情況 （1） 徐發(fā)仁主編?！稒C(jī)床夾具設(shè)計(jì)》。重慶大學(xué)出版社 （2） 林文煥 陳本通編著。《機(jī)床夾具設(shè)計(jì)》 國(guó)防工業(yè)出版社 （3） 王光斗 王春福等編。《機(jī)床夾具設(shè)計(jì)手冊(cè)》第三版 上海科學(xué)技術(shù)出版社 （4） 龔定安 蔡建國(guó)主編。《機(jī)床夾具設(shè)計(jì)原理》 陜西科技技術(shù)出版社 （5） 東北重型機(jī)械學(xué)院等編著?！稒C(jī)床夾具設(shè)計(jì)手冊(cè)》上海科學(xué)技術(shù)出版社 （6） 大連理工 王小華主編?！稒C(jī)床夾具圖冊(cè)》 機(jī)械工業(yè)出版社 （7） 羅家良主編 《機(jī)床夾具實(shí)例分析》 上海第一機(jī)電科技局科技情報(bào)研究所 五、指導(dǎo)教師審批意見(jiàn) 指導(dǎo)教師： （簽名） 年 月 日 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)（論文） 摘要 二十一世紀(jì)的制造業(yè)面臨著顧客需求驅(qū)動(dòng)、不可預(yù)測(cè)、快速多變和來(lái)自全球不斷增加的市場(chǎng)競(jìng)爭(zhēng)，而且競(jìng)爭(zhēng)不斷加劇。市場(chǎng)的不斷變化要求制造系統(tǒng)加工的產(chǎn)品品種能夠快速變換以滿足市場(chǎng)需求。近來(lái)的制造業(yè)發(fā)展表明，夾具能比較好的滿足上述要求并符合我國(guó)國(guó)情。作為制造系統(tǒng)重要組成部分的夾具設(shè)計(jì)部分，制造系統(tǒng)對(duì)其提出了新的要求。夾具在機(jī)械加工起著重要的作用，它直接影響著機(jī)械加工的質(zhì)量，生產(chǎn)效率和成本，因此夾具設(shè)計(jì)是機(jī)械工藝準(zhǔn)備和施工中的一項(xiàng)重要工作。 這篇畢業(yè)設(shè)計(jì)主要闡述的是一套關(guān)于電機(jī)座液壓夾緊粗鏜夾具的設(shè)計(jì)方法，這種夾具主要應(yīng)用于電機(jī)座的粗鏜工序，通過(guò)這個(gè)夾具可以保證電機(jī)座粗鏜后的加工要求和達(dá)到提高生產(chǎn)效率的目的。 在說(shuō)明書中，首先明確了設(shè)計(jì)任務(wù)并對(duì)夾具作了相關(guān)的闡述，接著根據(jù)電機(jī)座的加工工序提出了鏜床夾具的設(shè)計(jì)方法和理論，并依據(jù)這些方法和理論對(duì)夾具進(jìn)行設(shè)計(jì)和校核、驗(yàn)算。最后對(duì)夾具中某些典型或重要的零件進(jìn)行了介紹和校核。這種設(shè)計(jì)方法代表了鏜模的一般設(shè)計(jì)過(guò)程，對(duì)其他夾具的設(shè)計(jì)工作也有一定的價(jià)值。 關(guān)鍵詞：夾具、電機(jī)座、工序 Abstract Manufacturing companies in the 21s' century will face unpredictable, high-frequency market changes driven by global competition. The continuous changes of market require a rapid change of product varieties in order to meet the market demands. Recent surveys have showed that fixture are the cornerstones of this new manufacturing paradigm. Fixture is also suitable to our country. After reconfiguration manufacturing system puts new requirements on fixture design, an important component of the traditional manufacturing system. Fixture is very important equipment in process of machine manufacturing because it can directly affect the quality of products and productivity and cost. So fixture designing is also a base portion in machine process. This paper of graduation mainly presents a systematic approach for the design of Electrical Block hydraulic upholding tool rough noise of drums .this upholding tool are mainly used in the working procedure of Electrical Block hydraulic rough noise of drums . It can accomplished the purpose what this fixture can be satisfy after Electrical Block hydraulic upholding tool rough noise of drums produce demand and approach to improve production efficiency. First ,we define the design of assignment and represent something about fixture. The next ,Accordingly to the processing of workpiece of motor cabinet ,we extract the method and way of boring lathe of fixture. And then ,we practice into design the fixture according to the method and way .After this design ,we take some canonical workpiece for example to check the strength.. The method of this fixture represent the process of canonical fixture. Of course ,the design of our fixture also can guide the design of fixture. Keyword：fixture、motor cabinet、process 注： 3 河南理工大學(xué)畢業(yè)設(shè)計(jì)說(shuō)明書（2005屆） 目錄 前言………………….........................................................................................1 第1章 畢業(yè)設(shè)計(jì)的目的和任務(wù)……..………...…………………...2 1 畢業(yè)設(shè)計(jì)的目的………………………………………………………………2 2 畢業(yè)設(shè)計(jì)的任務(wù)……………………………………………….………………2 第2章 夾具方案設(shè)計(jì) 1 夾具的概述及作用………….………………………………………………4 1.1 夾具的概念…...............…..………………………………………………...4 1.2 夾具的作用………………………………...……………………………..…5 2 夾具設(shè)計(jì)的一般步驟 2.1 設(shè)計(jì)前的準(zhǔn)備工作……………………………………………………….…6 2.2 擬定夾具結(jié)構(gòu)方案、繪制工件工序圖………….………………………....7 2.3 確定夾具的結(jié)構(gòu)方案……………………………………………………….8 2.4 夾具體的設(shè)計(jì)………………………………………………………13 3 繪制夾具結(jié)構(gòu)草圖 3.1 布置圖面……………………………………………………………………15 3.2 設(shè)計(jì)定位夾緊元件…………………………………………………..……..16 3.3 夾具的精度計(jì)算……………………………………………………………18 3.4 夾緊力（矩）的驗(yàn)算……………………………….………………..…….19 3.5 繪制夾具總圖………………………………………………………...…….28 3.6 尺寸標(biāo)注與公差配合……………………………………………………....29 3.7 液壓傳動(dòng)原理…………………………………………………………...….32 3.8 夾具總體設(shè)計(jì)中須注意的問(wèn)題……………………………………..……33 第3章 典型零件介紹…………………………...………………..….44 總結(jié)……………………………………………………………..……………..48 致謝……………………………………………………………………………49 參考文獻(xiàn)…………………………………………………………………….50 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯（2006屆） 河南理工大學(xué) 本 科 畢 業(yè) 設(shè) 計(jì) (論 文) 英 文 翻 譯 院 (系部):機(jī)械與動(dòng)力工程學(xué)院 專業(yè)名稱:機(jī)械設(shè)計(jì)制造及其自動(dòng)化 年級(jí)班級(jí):2002級(jí)3班 學(xué)生姓名:李太平 指導(dǎo)教師:姜無(wú)疾 日 期: 2006年6月7日 Efficient prediction of workpiece-fixture contact forces using the rigid body model Michael Yu Wang1 , Diana M. Pelinescu2 1Department of Automation and Computer-Aided Engineering The Chinese University of Hong Kong, Shatin, N.T., Hong Kong 2Department of Mechanical Engineering University of Maryland, College Park, MD 20742, USA ABSTRACT Prediction of workpiece-fixture contact forces is important in fixture design since theydefine the fixture stability during clamping and strongly influence workpiece accuracyduring manufacturing. This paper presents a solution method for predicting the normal and frictional contact forces bet- ween workpiece-fixture contacts. The fixture and workpiece are consid- ered to be rigid bodies, and the model solution is solved as a constrained quadratic optimization by applying the minimum norm principle. The model reveals some intricate properties of the passive contact forces, including the potential of a locator release and the history dependency during a seq- uence of clamping and/or external force loading. Model predictions are shown to be in good agreement with known results of an elastic-contact model prediction and experimental measurements. This presented method is conceptually simple and computationally efficient. It is particularly useful in the early stages of fixture design and process planning. 1 INTRODUCTION Fixture design is a practical problem and is crucial to product manufac- turing.In particular, the positioning and form accuracy of the workpiece being machined might be highly influenced by the contact forces between the work- piece and the fixture elements of locators and clamps. Localized contact forces cancause elastic /plastic deformation of the workpiece at the contact regions. This can contribute heavily to workpiece displacement and surface marring. On the other hand, insufficient contact forces may lead toslippage. This research work is supported in part by the US National Science Foundation (grants DMI- 9696071 and DMI-9696086), the ALCOA Technical Center (USA), the Hong Kong Research Grants Council (Earmarked Grant CUHK4217/01E), the Chin- ese University of Hong Kong (Direct Re-search Grant 2050254), the Ministry of Education of China (a Visiting Scholar Grant at the Sate Key Laboratory of Manufacturing Systems in Xi’an Jiaotong University), and the Natural Science Foundation of China (NSFC) (Young Overseas Investigator Collaboration Aw- ard 50128503) or separation of the workpiece from a locator during the manu- facturing process. Frictional forces at the workpiece-fixture contacts may help prevent workpiece from slipping and therefore act as holding forces. Their presence, however, increases the complexity of fixture analysis and design. Therefore, it is of significant help to provide the fixture designers with good knowledge of the contact forces based on an efficient engineering analysis. This would al-low the designers to be able to determine the best fixturing scheme that would minimize product quality error [1]. The essential requirement of fixturing concerns with the kinematic concepts of localization and force closure, which have been extensively studied in recent years. There are several formal methods for fixturekinematic analysis based on the assumptions of rigid workpiece and fixture and frictionless workpiece fixture contacts [2, 3]. Conventional fixture design procedures have been described in traditional design manuals [4], while feature-based, geometric- reasoning, or heuristic approaches have also been employed in automated fixture design schemes [5, 6, 7]. For the analysis of workpiece-fixture contact forces a comprehensive approach is to consider the workpiece-fixture system as an elastic system. This system can be analyzed with a finite element model [8, 9, 10, 11]. Such a model is often sensitive to the boundary conditions. It also results in a large sizemodel and requires high computational effort. Thus, this approach is not suited for the early stages of design of fixture layout and clamping schemes. The modeling complexity may be reduced if quasi-static loading conditions are assumed and a local elastic/ plastic contact model is used at each workpiece- fixture contact [12, 13]. In using the principle of minimum total complementary energy [14], the geometric compatibility of workpiece-fixturedeformation is maintained without resorting to any empirical force-deformation relation such as the meta-functions used in [15].system usually is statically indetermin- ate,especially in the the presence of friction [16, 9]. It is not unusual in the literature that the frictional forces are ignored so that the issue of static indeterminacy is avoided, in spite of the significant impact that the frictional forces can make. In this paper we present a solution method for theprediction of workpiece- fixture contact forces based onthe rigid body model and Coulomb friction model. The method is based on the application of the minimumnorm principle with frictional forces as constraints. As a result, it yields a unique solution for the contact forces without requiring computationally intensive numerical procedures. The paper focuses on two areasof discussions contributing to the general understanding of workpiece fixturing. (1) It is shown that the minimum norm solution of the workpiecefixture contact system can be regarded as a special form of the minimum energy principle. The proposed method gives a quick estimate of the contact forces without the need of a deformation model of the workpiece-fixture contact. When compared with experiment data and results of another approach, the prediction accuracy of the rigid body model approach is considered reasonable. This indicates that the proposed method might be particularly useful in the early stages of fixture layout and clamping scheme design. (2) The second focused discussion of the paper is the concept of history dependency of the frictional contact forces. The fixture contact forces are considered reactive forces to applied forces on the workpiece. When a friction constraint is active as defined by Coulomb’s law, the minimum norm solution reveals that the reactive frictional contact forces will depend on the sequence in which the external and/or clamping forces are applied on the workpiece. This history dependency may have a strong implication in work piece clamping especially when multiple clamps are applied. 2 THE CONTACT SYSTEM MODEL 2.1 Fixture elements For the purpose of analysis of workpiece-contact forces in this paper, the basic elements of a fixture are classified into passive and active types as locators and clamps. Here, a locator is referred to as a component to provide a kinematic constraint (position and/or rotation) on the workpiece. A locators represents a passive element. It includes the conventional locator pins or buttons that are used essential for a unique localization of the workpiece withrespect to a fixture reference frame. A support of a movable anvil that is sometimes used for providing additional rigidity to the workpiece is also treated as a locator for the purpose. A support is usually actuated by spring force (pop-up support), screw thread (jack support), or by hydraulics. In all cases, it is engaged only after workpiece localization and is locked into place once it makes contact with the workpiece, transforming it into a passive element. A clamp is represented as a force applied on the workpiece to provide a complete restraint of the workpiece against any external forces on the workpiece. Clamps are typically engaged manually or pneumatically. Clamping forces are said to be active elements, so as the external forces. These fixture elements are illust- rated in Fig. 1. 2.2 Frictional contact Within the framework of rigid body model, we describe each workpiece-fixture contact with a point contact model with Coulomb friction for clarity [2, 3]. As shown in Fig. 2, the frictional contact produces three force components on the workpiece, with force intensities ( z , x , y ) for the normal and tangent directions respectively. Here, the inward surface unit normal of the workpiece is represented by n , while t and b represent two orthogonal tangent unit vectors. The tangential forces are due to friction as defined by Coulomb’s law. For a locator i contacting the workpiece at position i , the contact force and moment exerted on the workpiece is represented as where Clamps are also defined similarly as point contacts. A clamp j is located at rj along the surface unit normal nj . It also exerts force and moment on the workpiece. However, the normal and the tangential clamping forces are considered in a different way. The normal clamping force is an active force and is treated as given. The tangential clamping forces are frictional forces that usually cannot be controlled in clamp actuation. They may have to be considered as unknowns and to be solved for. Thus, the clamping force and moment exerted on the workpiece is given as where denotes the clamping force intensity ( >0), and hn , j, ht , j and hb , j are also defined accordingly. 2.3 Coulomb’s friction law A simple Coulomb’s friction law is applied to the tangential forces such that for every locator contact and clamp contact respectively with corresponding friction coefficients and . 2.4 The force equations Suppose that the fixture has n locators and m clamps. Let Q represent all external force (and its moment) vectors applied on the workpiece. Then, the static equilibrium equation of the workpiece is given as Withindicating the intensity vector of the unknown passive forces at all contacts. 3 THE METHOD OF MINIMUM NORM SOLUTION 3.1 The minimum norm principle For a general three-dimensional workpiece its fixture would must have at least 6 locators and one clamp, i.e., n 6 and m 1. In the presence of friction, the fixture system represented by Eq.11 is statically undeterminate. If _ clamps are simultaneously applied, there exist (3n+2m) unknown intensities of the reaction forces at all locator and clamp contacts in the equilibrium equation (Eq. 11). Within the framework of the rigid body model the workpiece -fixture contact problem is solved by invoking the principle of minimum norm [17]. This principle essentially states that of all possible equilibrium forces for a rigid body subjected to prescribed loading, the unique force solution compatible to the equilibrium renders a minimum force norm. This is mathematically described as Thus, the contact force solution is represented by a quadratic minimization with equality and inequality constraints. The linear equality constraints of Eq.15 describe the equilibrium state. The inequality constraints of Eq.16 maintain that the workpiece fixture contacts are passive and unilateral, while Eq.17 and Eq.18 define the tangential forces to obey Coulomb’s friction law. In addition, it is required that so the clamping forces are applied always inward to the workpiece. It should be pointed out that the minimum norm principle is equivalent to the principle of minimum complementary energy for an elastic contact system [13, 14], if we consider it to be linear and with contact elasticity defined by a compliance matrix W. In that case, the complementary energy is defined by . Thus, the minimum norm principle provides a solution in a similar sense but under the simpler provision of rigid body contact. 3.2 Solution procedures A standard optimization routine may be used for the numerical solution of Eq.14 as a quadratic minimization with linear equality constraints and nonlinear inequality constraints, for example, the popular MATLAB system. For a typical case of practice, e.g., n=6 and m=1, it is usually takes less than a few seconds to obtain a solution on a common 1GHz PC. Another numerical approach, as often used in a robotic grasping analysis [18], is to approximate the friction cones of the nonlinear inequality constraints Eq.17 and Eq.18) with polyhedral convex cones [19]. This will replace the nonlinear constraints with a number of linear ones. The polyhedral approximation of the friction cones results in a minimum norm solution system with linear equality constraints and lower bounds on variable. Thus, a standard quadratic programming method could be used for efficient solution. Typically, it is sufficient to use a 4-12 sided polyhedra for an sufficiently accurate result [13, 19]. Practically, this appro- ximation method does not offer significant computational advantage since the number of locators and clamps in an industrial fixture is relatively small, typically in a total of 7-12. 4 CONTACT FORCES IN CLAMPING Eq.14 deals with a general case of multiple loads of clamping and external forces applied on the workpiece simultaneously. Under the unilateral and/or frictional inequality constraints, the minimum norm principle would reveal a number of intricate properties of the solution. For conceptual clarity we shall first examine the case of a single clamp in the fixture and without any external loads, i.e., m=1 and Q=0. A understanding of the special properties is essential for obtaining a complete solution for the general workpiece-fixture system. In particular, the following situations are examined: (1) the minimum-norm generalized inverse solution, (2) internal contact forces, (3) a locator release, (4) frictional forces at the clamp, and (5) the potential of history dependency of the contact forces. 4.1 The specific solution When the workpiece is considered to subject to a single clamp only m=1 and Q=0, the equilibrium equations become If all locators generate reactive forces and all frictional forces of the locators and the clamp are within their respective friction cones, i.e., and , for the clamp, then it is said that all the inequality constraints are inactive. In this case, the minimum norm solution for Eq.19 is easily obtained as directly in terms of the minimum-norm generalized inverse of matrix ,which is also known as the left pseudo-inverse [17]. This is the specific solution to the linear system (Eq.19), which is effectively unconstrained. It is well known that the unconstrained linear system attains its minimum norm with the specific solution and its homogeneous solution vanishes [17]. The system of contact forces is essential linear in this case where at each contact its normal contact force exists and its friction forces lie strictly inside the friction cone. From an optimization point of view, it can be said that the solution satisfies the Kuhn-Tucker (K-T) conditions as a minimum point. 4.2 Internal contact forces However, when any of the locators becomes nonreactive (i.e., zi=0) and/or the limit friction is reached at a locator or the clamp, one or more inequality constraints become active. Then, the solution to Eq.19 with all relevant constraints has to be solved as a minimum norm solution [17],i.e., min// a// , with a numerical procedure as described above. So the minimum-norm solution is in the form of (21) The first term is the specific solution of Eq.20, and the the second term is said to be the homogeneous solution. According to the linear algebra, the specific solution is a projection of the minimum-norm solution defined as by the projection matrix . The homogenous solution _ is the other orthogonal projection given as Thus, in using the common terminology of robotics, the homogenous component shall be referred to as theinternal forces among the locators and clamps. In reaction to the clamping force represented by , the specific solution component is generated at the contacts to balance the clamping force only, while the homogenous solution component is to solely maintain the unilateral and frictional contact constraints. The constraint satisfaction is achieved at the cost of increasing the contact force intensities. Internal forces in the fixture are passive forces as a result of a reaction to the applied load, unlike those of a multi-fingered hand which could be actively controlled and arbitrarily specified. 4.3 Locator release It is possible that the minimum-norm principle yields a solution with contact forces to vanish at a locator, i.e. . This situation is called locator release, since this locator does not generate any reaction forces to the given load. In the presence of friction, this is especially possible, even in the case of minimally required kinematic localization of six locators. In other words, a clamp or an external load may render one or more locators to release, creating a potential situation of locator lift-off. These situations are undesirable in practice. 4.4 Frictional forces at the clamp In Eq.19 the unknown contact forces include the frictional forces at the clamp contact. In the case that the only loading is from this clamp itself with its normal force , the frictional forces at the clamp would not exist, i.e.,. This is evident from fact that the contact normal is orthogonal to the contact tangent plane, or .A clamp cannot generate friction forces for itself. However, friction forces could be generated by other clamping or external forces. This is related to the issue of history dependency of contact forces discussed next. 5 HISTORY DEPENDENCY OF FRICTIONAL FORCES\ 5.1 Sequential loadings in fixture From an operation point of view workpiece fixturing may have five basic steps : (1) stable workpiece resting under gravity, (2) accurate localization, (3) support reinforcement, (4) stable clamping, and (5) external force application. These steps have strong precedence conditions. When a workpiece is placed into a fixture, it must first assume a stable resting against the gravity. Then, the locators should provide accurate localization. Next, support anvils (if any) are moved in place, and finally clamps are activated for the part (or force-closure) immobilization. The part location must be maintained in the process of instant- tiating clamps without workpiece lift-off. 5.2 Loading history and pre-loads As discussed in Section 4.2, an instantiating clamping load or external load may render an inequality constraint active and cause internal forces among the contacts. The (Eq.14) equilibrium system becomes nonlinear. Thus, the linear superposition principle would not apply for this load with any other clamping or external load that is applied at another time. The contact forces reactive to this load will become preload forces for the contacts when another load is applied later. In other words, the contact force solution for an instantiating (clamping and/or external) load depends on the contact forces that are already in existence. The contact forces may depend on their history. When the potential of history dependency is considered, the contact force system of Eq.14 should be described more precisely as follows. Let’s denote the existing contact forces by _ and the next applied load is a clamping load , an external load , or both if they are applied simultaneously. The contact forces a in response to this instantiating load only would satisfy The resultant contact forces of all the sequential loadings are given as Thus, the total contact forces may depend on the specific sequence in which the clamps and external forces are loaded on the workpiece. Practically, hydraulic or pneumatic clamps may provide for simultaneous clamping, while manual clamps are generally loaded individually. Considering the potential of history dependency (or sequence dependency), even when simultaneous c