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十、三拐曲轴 见图2-181、零件图样分析l）55mm两轴径同轴度公差为0.03mm。2）1:20锥度部分对A-B轴心线同轴度公差为0.03mm。3）三个拐径分别对A-B轴心线平行度公差为0.03mm。4）热处理，人工时效处理227270HBS。5）曲轴材料为QT600-3。2、三拐曲轴机械加工工艺过程卡 (表2-10)表2-10三拐曲轴机械加工工艺过程卡工序号工序名称工序内容工艺装备1铸铸造（左端55mm处铸造尺寸为75mm）2清砂清砂3热处理人工时效处理4清砂细清砂5涂漆非加工表面涂红色防锈漆6划线按毛坯外形找正，照顾各加工面，划外形尺寸线7铣经两轴径部分定位压紧分别铣两个端面保证总长尺寸520mm，钻左端中心孔B4X62W（端铣）8粗车夹右端外圆，找正两轴径外圆。顶左端中心孔，车两轴径处，其中1：20一端（右端）尺寸为62mm，另一端（左端）尺寸为70mm（工艺尺寸）CW61639粗车倒头装夹工件左端70mm处，中心架夹带锥一端62mm轴径上，钻右端中心孔B4，粗车锥度一端各部尺寸，留加工余量5mm（其中70mm车至图样尺寸）CW616310划线在左端70mm轴径上划键槽线深5mm、宽10mm、长35mm（工艺用键槽），注意与靠70mm最近的拐在同一平面内11铣以两70mm定位装夹工件铣键槽5mm10mm35mmX52K12粗车采用专用工装装病例工件粗车曲轴三个拐径及拐径两个侧面，（专用工装为转夹具，可进行三等分分度）留加工余量5mm专用工装CW616313精车采用专用工装装夹工件精车曲轴三个拐径及拐径两个侧面，留磨量0.8mm1mmCW616314精车夹工件左端，顶右端中心孔，车工件右端各部尺寸，留加工余量0.81mm，车1：20锥度留加工余量1mmCW616315精车倒头，采用两顶尖装夹工件，车左端尺寸55mm至55mm，倒角R5CW616316检验检查曲轴偏心距17磨以两中心孔定位装夹工件（专用工装），磨拐径三处至图样尺寸55mm，靠磨拐径两侧及圆角R5M824018磨以两中心孔定位装夹工件，磨轴径两处55mm至图样尺寸磨50mm至图样尺寸M1432A19磨夹工件左端，中心架夹右端55mm处，找正，磨1：20锥度至图样尺寸M1432A锥度环规铜皮20划线划12mm6mm键槽21铣以两轴径定位装夹工件铣12mm6mm键槽X52K22钳修锉飞刺23检验磁粉探伤24检验按图样检验工件各部尺寸精度25入库涂油入库3、工艺分析 1）该工件为三拐曲轴，其形状复杂，加工技术要求较高。为加工三拐径，应制做专用工装 (通常工厂称为分度转夹具)，其要求为，能够均分三等份 (曲轴三拐径偏心距为60mm)，并要保证回转平衡。 2）工件加工时应将粗、精车分开，必要时也可以将粗、精磨分开，这样有利于工件加工精度。 3）为了使用回转分度夹具加工曲轴的三个拐径，在铸造工序上有意加大曲轴左端轴径尺寸，留出工艺键槽的加工余量。4）所有轴径及轴径线上的各部尺寸的加工均以两中心孔为定位基准，所以同轴度均由设备 (工艺)来保证。如需要检查两轴径同轴度，可以用一对标准的V形块支撑两个55mm铀径，用百分表测量。 5）工件1:20锥度的检查采用专用环规检查。 6）曲轴偏距的检测可以参照单揭曲轴偏心距的检测方法进行检测。 7）三个拐径120均布的检查，可参看图2-19，用一对标准V形块 (V形块安放在标准平台上)，支撑工件两端轴径55mm，然后调整两端便支撑处轴径的轴心线与平台平行。图2-19 三拐曲轴120等分检测示意图 用高度尺测量尺寸AA，使拐径中心和轴径中心连接在水平面夹角为30，通过计算算出拐径角度误差。 AA尺寸的计算方法： AA=BB-=BB-+ 式中 AA标准30位置拐径计算的理论值（各辆径、拐径尺寸均为实测值）（mm）； BB支承轴径外圆实际高度（mm）； De曲轴轴径实际尺寸（mm）； D曲轴拐径实际尺寸（mm）； E偏心距（mm）。 按计算出AA尺寸调整好一侧拐径位置之后，测量与之相对应的拐径高度值B，若B值与AA值相等，即等分合格；若B值与AA值不相等，这时应计算出拐径中心和轴径中心连线与水平面的夹角： sin=(见图2-19) m= n=B-(BB-)=B-BB+m=-(B-BB+)=-B+BB-sin=(-B+BB-)/E式中拐径中心和支承轴中心连线与水平面的夹角； B实际测量尺寸；m、n为计算值而给出的中间变量。=30-如果为负值，则此拐径与标准30位置拐径夹角小于120，反之夹角大于120。8） 拐径轴线对轴径的轴线平行度误差，可在图2-19的基础上测出拐径外圆最高点处最长距离的两点差值，即为两轴线的平行度误差，三个拐径分别测量即可。夹具总装配图1份，主要零件如夹具体、钻模板、定位销等零件图至少3张，设计说明书等。5宁波大红鹰学院毕业设计（论文）任务书所在学院机械与电气工程学院专业机械设计制造及其自动化班级11机自2学生姓名王泽华学号1121080333指导教师杨豫新题 目CA-10型汽油机曲轴加工工艺规程及夹具设计本课题是学完了专业课程，应用所学的基础理论、专业知识与技能去分析和解决生产实际问题的一个综合锻炼。通过这个毕业设计，巩固自己所学的理论知识与技能，提高自己的设计、计算、绘图、技术文件编写等各方面的能力。学会正确使用技术资料、标准、手册等工具书。并在完成毕业设计的过程中，培养自己理论联系实际、严肃认真的工作作风和独立分析，独立思考的能力。一、 毕业设计（论文）工作内容与基本要求1、主要任务与目标（1）围绕课题，查阅相关文献资料不少于12篇，其中外文资料不少于2篇，并撰写文献综述，字数不少于3000字。（2）完成与课题相关外文文献（原文论文）的翻译，字数不少于2000字；（3）分析和研究课题内容，查阅文献资料，完成开题报告； （4）完成CA-10型汽油机曲轴零件图绘制和毛坯图设计；完成CA-10型汽油机曲轴的机械加工工艺规程（含机械加工工艺过程卡片、机械加工工序卡片）设计；数控加工需手工或自动编制数控加工程序；完成2套专用夹具（含总装图、夹具零件）等零件的设计，编写设计说明书1万字以上。（5）完成毕业设计的其它工作（完成指导记录、资料上传网络设计平台、答辩PPT准备、资料装订整理）。2、主要内容与基本要求（1）主要内容：1）了解轴类零件功能、结构和工艺特点、技术要求，毛坯特点；2）完成CA-10型汽油机曲轴机械加工工艺规程及夹具的设计。（2）要求：1）机械加工工艺设计方案合理；2）要求夹具结构合理、紧凑，使用方便；3）完成3张A0图纸（折合），并要求CAD绘制；毛坯图一张、夹具总装图、夹具主要部件图和零件图；4）图纸、图面布置合理、正确清晰符合制图标准及有关规定。5）撰写设计说明书内容包括：课题的目的、意义、国内外动态；研究的主要内容；机械加工工艺路线和夹具结构方案的拟定和主要参数的设计计算；加工工艺和夹具方案的确定，设计专用夹具，进行计算，条理清楚，计算有据；格式按宁波大红鹰学院机电学院机械设计制造及其自动化专业全日制普通本科生毕业论文（设计）规范化要求。文字在1万字以上。6）其他要求：在进行设计过程中，不能照抄现有资料，应当经过自己的努力，理解和消化资料，认真分析和解决实践过程中遇到的问题，充分发挥自己独立思考和创作设计，培养和锻炼工程实际中的发现问题和解决问题的能力，反对盲从和抄袭行为。设计期间应完成以下工作：查阅相关文献资料，其中外文资料不少于两篇，外文翻译不低于2000字，英文翻译必须注明出处，并提供原版PDF文件。3、设计参数（1）年产量为50000台/年；零件技术要求见图纸；（2）产品要求优质、高产、低消耗。4、应收集的资料及主要参考文献应收集的资料：箱体的结构、材料、功用及相关技术要求；箱体的工艺性；工艺路线拟定的方法；各种专用夹具的种类、特点及原理，钻床、铣床、数控机床类夹具的设计方法及国家机械制图标准。主要参考文献：1 尹成湖机械可制造技术基础M.北京：高等教育出版社，2008.82 李凯岭,宋强机械制造技术基础M.济南：山东科学技术出版社，2005.93 李凯岭机械制造技术基础M.北京：科学出版社，20074 尹成湖 李保章 杜金萍.机械制造技术基础课程设计M.高等教育出版社，20095 柯建宏.机械制造技术基础课程设计M.武汉：华中科技大学出版社，2008.86 浦林祥.金属切削机床与夹具金属切削机床夹具设计手册M.机械工业出版社1995第2版7 张滢滢，刘春雨.关于机床夹具的发展前景的探讨J. 科技创新导报 2011年26期 8 陈宏钧.实用机械加工工艺手册M.机械工业出版社，1997.69 陈洪涛.数控加工工艺与编程M.高等教育出版社，2009.1210 吴拓.简明机床夹具设计手册M. 化学工业出版社，2010.411 毛平淮.互换性与测量技术基础：第2版M.机械工业出版社，2011.1二、毕业论文进度计划序号各阶段工作内容起讫日期备注1阅读毕业设计任务书，明确设计的要求和任务，收集并阅读文献资料，完成开题报告，文献综述和外文翻译文献。2014.10.111.30外文资料不少于两篇，翻译不低于2000字，提供原版PDF。2分析ST型减速器主机座箱体零件图，确定毛坯类型及制造方法，完成毛坯图。10.910.203进行选择基准，拟定机械加工工艺路线10.2110.314确定机械加工余量、工序尺寸及公差，选择机床及工艺设备11.111.125确定切削用量及基本工时，并填写工艺文件11.1311.306确定专用夹具工件的定位方案和夹紧方案，计算定位误差和计算夹紧力，绘制夹具总图和零件图12.0112.311230中期检查7撰写毕业设计说明书。整理毕业设计资料并上交，准备答辩。201501010331资料装订完善、上交8答辩58三、专业（教研室）审批意见：审批人（签字）：毕业设计文献综述的写作要求 一、撰写文献综述的基本要求文献综述是针对某一研究领域或专题搜集大量文献资料的基础上，就国内外在该领域或专题的主要研究成果、最新进展、研究动态、前沿问题等进行综合分析而写成的、能比较全面的反映相关领域或专题历史背景、前人工作、争论焦点、研究现状和发展前景等内容的综述性文章。“综”是要求对文献资料进行综合分析、归纳整理，使材料更精练明确、更有逻辑层次；“述”就是要求对综合整理后的文献进行比较专门的、全面的、深入的、系统的评述。二、撰写文献综述的基本注意事项1、文献综述是一篇相对独立的综述性学术报告，包括题目、前言、正文、总结等几个部分。题目：一般应直接采用文献综述作为标题，经指导教师批准也可以所研究题目或主要论题加“文献综述”的方式作为标题。前言：点明毕业论文（设计）的论题、学术意义以及其与所阅读文献的关系，简要说明文献收集的目的、重点、时空范围、文献种类、核心刊物等方面的内容。正文：无固定格式，文献综述在逻辑上要合理，可以按文献与毕业论文（设计）主题的关系由远而近进行综述，也可以按年代顺序综述，也可按不同的问题进行综述，还可按不同的观点进行比较综述。总之要根据毕业论文（设计）的具体情况撰写，对毕业论文（设计）所采用的全部参考文献分类、归纳、分析、比较、评述，应特别注意对主流、权威文献学术成果的引用和评述，注意发现已有成果的不足。结论：对全文的评述做出简明扼要的总结，重点说明对毕业论文（设计）具有启示、借鉴或作为毕业论文（设计）重要论述依据的相关文献已有成果的学术意义、应用价值和不足，提出自己的研究目标。2要围绕毕业论文主题对文献的各种观点作比较分析，不要教科书式地将与研究课题有关的理论和学派观点简要地汇总陈述一遍。3评述（特别是批评前人不足）时，要引用原作者的原文（防止对原作者论点的误解），不要贬低别人抬高自己，不能从二手材料来判定原作者的“错误”。4文献综述结果要说清前人工作的不足，衬托出作进一步研究的必要性和理论价值。5采用了文献中的观点和内容应注明来源，模型、图表、数据应注明出处，不要含糊不清。6文献综述最后要有简要总结，并能准确地反映主题内容，表明前人为该领域研究打下的工作基础。7所有提到的参考文献都应和所毕业论文（设计）研究问题直接相关。8文献综述所用的文献，与毕业设计（论文）的论题直接相关，与毕业论文（设计）的参考文献数量完全一致；重要论点、论据不得以教材、非学术性文献、未发表文献作为参考文献，应主要选自学术期刊或学术会议的文章，其次是教科书或其他书籍。至于大众传播媒介如报纸、广播、通俗杂志中的文章，一些数据、事实可以引用，但其中的观点不能作为论证问题的依据。三、撰写文献综述的其他事项1一篇毕业论文应完成一篇文献综述，字数规定不少于3000字。2文献综述所用的文献，至少要求12篇，且外文的至少要求2篇。3文献综述应包括综述题目、综述正文、文献资料等几方面内容。4文献综述所引用的参考文献书写格式应符合GB77141987参考文献著录规则。常用参考文献的书写格式如下：1.专著： 序号作者.书名M.版本(第1版不著录).出版地:出版者,出版年.起止页码.2.期刊: 序号作者.题名J.刊名,年,卷(期):起止页码.3.会议论文集（或汇编）: 序号作者.题名A.编者.论文集名C.出版地:出版者,出版年.起止页码.4.学位论文: 序号作者. 题名D. 学位授予地址：学位授予单位，年份.5.专利: 序号专利申请者. 专利题名P.专利国别（或地区）:专利号, 出版日期.6.科技报告: 序号著者. 报告题名R.编号，出版地：出版者，出版年.起止页码.7.标准: 序号 标准编号，标准名称S.颁布日期.8.报纸文章 : 序号 作者. 题名N. 报纸名，年-月-日（版次）.9.电子文献: 序号 主要责任者.电子文献题名电子文献及载体类型标识.电子文献的出处或可获得地址，发表或更新日期/引用日期(任选).10.各种未定义类型的文献: 序号主要责任者.文献题名Z. 出版地：出版者，出版年.文献综述范文：文献综述前言本篇毕业设计（论文）题目是从供应链的角度浅谈光伏产业的问题与对策。太阳能光伏产业作为清洁能源的主要来源，已经越来越受到政府、企业、研究机构乃至个人的重视，为了发现光伏产业在现代中国的利用意义和价值，实现人与自然的可持续发展，本文着力研究光伏产业供应链的两端：上游供应链即多晶硅的采购，下游供应链即太阳能光伏产品的销售物流。面对光伏产业“两头在外”的尴尬局面，即原料的采购和产品的销售很大程度上依存于国外市场，本文从供应链的角度，利用图书资料、互联网信息、企业调查等方法，探求中国光伏企业的未来发展方向，最后又以南京中电集团为例，简述该企业在光伏行业中未来的发展之路。正文能源的紧缺，已经成为目前世界范围内的热门话题。从长远战略上考虑，开发和利用太阳能成为了各国可持续发展战略的重要组成部分。太阳能是一种既丰富又无污染的可再生能源，通过太阳能电池将光能转化成电能，也就是我们所说的太阳能光伏产业。世界各国都对太阳能光伏产业给予了前所未有的重视。国家发改委能源研究所专家预测，我国太阳能电池装机容量的年增长率有望超过40%。到2020年，系统年产值将接近3000亿元。与此相应，我国光伏产业必然会有极大的发展空间。1、光伏产业供应链的基本情况整个光伏产业主要包括多晶硅原料、太阳能电池、集成组件、发电工程四个相关的行业。供应链主要包括硅材料（主要是多晶硅）的生产和供应，电池片制造组件系统封装与应用，光伏产品的分销。其中，进入壁垒最高的环节为太阳能级晶体硅的生产，由于其技术与工艺上的难度，目前基本被国际上7大厂家垄断，这属于产业链上游环节。多晶硅的需求主要来自于半导体和太阳能电池。按纯度要求不同，分为电子级和太阳能级。其中，用于电子级多晶硅占55左右，太阳能级多晶硅占45，随着光伏产业的迅猛发展，太阳能电池对多晶硅需求量的增长速度高于半导体多晶硅的发展，预计从2008年，太阳能多晶硅的需求量将超过电子级多晶硅。 2、光伏产业发展趋势由于光伏产业巨大的发展前景以及现阶段硅材料供不应求引起的价格大幅上涨，许多投资者和政府、企业对该行业表示出极大的兴趣。资料显示，2005年，每公斤多晶硅价格仅为66美元，到2007年12月，已上升为每公斤400美元的天价。在刚刚过去的2007年，多晶硅引起了市场的充分关注。在市场缺口加大、价格不断上扬的刺激下，国内又一次涌现出投资多晶硅项目热潮。在这股热潮下，一部分专家指出，中国面临着大量产能上马将逐渐填补供给缺口，但未来可能存在过剩的风险；还有一部分专家则认为，过剩节点难以预测拐点 最终取决未来太阳能需求。但是不容忽视的是，某些地区不顾当地的实际情况，追逐眼前利益，有盲目上马多晶硅项目的趋势。3、光伏产业的从供应链角度所存在的问题概括来说，是以下3种情况。上游供应链卖方市场，中游供应链两头在外，下游供应链买方市场。具体来说，国内多晶硅生产企业在产业化方面的差距主要表现在以下几个方面： 3.1 产能低，上游和中游供需矛盾突出。2005年中国太阳能用单晶硅企业开工率在2030，半导体用单晶硅企业开工率在8090，无法实现满负荷生产，多晶硅技术和市场仍牢牢掌握在美、日、德国的少数几个生产厂商中，严重制约我国产业发展。 3.2 上游硅材料企业生产规模小。现在公认的最小经济规模为1000吨/年，最佳经济规模在2500吨/年，而我国现阶段多晶硅生产企业离此规模仍有较大的距离。工艺设备落后，同类产品物料和电力消耗过大，三废问题多，与国际水平相比，国内多晶硅生产物耗能耗高出1倍以上，产品成本缺乏竞争力。 3.3 行业供应链及地区供应链不够健全。地方政府和企业项目投资多晶硅项目，存在低水平重复建设的隐忧。3.4 下游市场在外。技术和市场被国外控制，存在“两头在外”的尴尬局面。中国光伏产业采购成本高，销售利润薄，有趋向于“代工生产”的趋势。4、如何解决问题 4.1 发展壮大我国多晶硅产业的市场条件已经基本具备、时机已经成熟，国家相关部门加大对多晶硅产业技术研发，科技创新、工艺完善、项目建设的支持力度，抓住有利时机发展壮大我国的多晶硅产业。 4.2 政府主管部门加强宏观调控与行业管理，避免低水平项目的重复投资建设，保证产业的有序、可持续发展。4.3 做大做强核心企业，完善产业链，形成产业集群。反对不计成本的引进和仿制，打破行业垄断，鼓励民营资本和国外资本进入，实现资源的合理配置。4.4 风险规避。产业链上下游企业结盟，阻止打价格战，实现供应链的良性循环。 4.5 支持最具条件的改良西门子法共性技术的实施，加快突破千吨级多晶硅产业化关键技术，形成从材料生产工艺、装备、自动控制、回收循环利用的多晶硅产业化生产线，材料性能接近国际同类产品指标；建成节能、低耗、环保、循环、经济的多晶硅材料生产体系，提高我们多晶硅在国际上的竞争力。 4.6 依托高校以及研究院所，加强新一代低成本工艺技术基础性及前瞻性研究，建立低成本太阳能及多晶硅研究开发的知识及技术创新体系，获得具有自主知识产权的生产工艺和技术。 4.7 以建设绿色奥运为背景，大力倡导环保节能产品的开发和利用。从下游供应链角度解决中国光伏产品的市场在外的问题，摆脱OEM的嫌疑，变中国制造为中国创造。5、中电集团在光伏供应链上所处的地位 中电集团与2006年成立了南京光伏公司，主要从事太阳能电池的生产和应用研究，集团下还有南京半导体材料公司、南京绝缘材料公司等，并计划在江西设立光伏的上游公司，主要从事多晶硅的生产。可以说，中电集团涉及了光伏行业的整个产业链，从上游的硅原料的生产到下游的太阳能电池片的生产和销售。6、中电光伏的优势和劣势61优势：611硅材料公司建成后，将控制了目前制约光伏行业发展最大的因素，即多晶硅高昂的价格。光伏企业可以以更低的成本进行生产，从而获得竞争力。612鉴于对整个产业发展前景看好的前提，加上近年硅原料短缺的实际状况造成产业倒置漏斗状布局，获得良好发展机会的重心向上游偏移。参与上游硅原料生产的公司短期内依然可以获得较为满意的利润。613对于上游硅材料公司的大量进入从而可能引起的产能过剩也不用过分担心。即便硅材料采购成本下降，亦可用于本集团内部光伏企业的生产。62劣势： 鉴于中国多晶硅生产技术的落后，硅材料生产成本高、耗能大、无法从根本上解决“三污”，从可持续发展的角度，作为一个由社会责任的企业，这些都应该是要考虑的问题。7、中电集团如何在光伏供应链中取得优势7.1健全产业网络组织。行业内部应建立产业联盟，行业外部面向上游要选择供应商，面向下游要面向市场和客户。7.2核心化生产。注重研发，生产核心产品，避免业务范围太广、太滥。7.3构建动态学习型组织。智力全球化，从全球范围内引进技术型和管理型专业人才。7.4资本全球化。采用中外合资或上市等手段，在全球范围内募集资金。7.5市场全球化。与国外采购商和销售商合作，以解决产品销售的后顾之忧。7.6早日投产。竞争将更为激烈。因此，促成硅料企业的早日投产，才是企业生存和制胜的关键。结论对于光伏产业的新进入者而言，好时代已经过去。在没有充足资金和稳定的供应渠道的前提下，光伏行业的生存是艰难的。因为在市场需求和原材料供应都是九成以上掌握在国外的时候，一个没有产业链角色和国际身份的企业，困难可想而知。经济全球化时代的这种背景下，游戏规则和制胜的核心就是全球配置资源。中国光伏产业应在全球范围内，发挥自身优势，寻求供应链上下游企业的良好合作与竞争。同时，政府应加强引导力度，大力提供金融和技术上的支持。参考文献1比尔罗伯茨,2006：硅与太阳能：马与马车，Electronic Business China，第10期。2陈学森，2007：国内多晶硅工业现状及相关发展政策建议，世界有色金属第10期3傅桂林、袁水林，2007：物流成本管理，中国物资出版社4何颖、赵争鸣，2004：新疆光伏发电现状及发展对策，新能源及工艺第4期5李红波、俞善庆，2006：太阳能光伏技术及产业发展，上海电力第4期6梁骏吾，2006：光伏产业面临多晶硅瓶颈及对策，科技导报第06期7王凤彬，2006：供应链网络组织与竞争优势，中国人民大学出版社8王文静,2006：硅材料紧缺对于太阳电池产业链影响的深度分析,中国科技成果杂志,第17期。9魏奎先等，2006：大阳能电池硅转换材料现状及发展趋势，轻金属第2期10郭瑾、李积和，2007：国内外多晶硅工业现状，上海有色金属3月第1期11肖户卫，2006：中国硅太阳能电池投资过热现象，中国科技信息第17期12杨国鑫、郑永孝，2007：多晶硅产业现状与发展趋势分析，煤16卷第3期13Adolf Goetzberger, Christopher llehliy.Photovoltaic materials,past,present,future. Solar Energy Materials&Solar Cells,62(2000):1-1914K.morita,T.MiKi.Thennodynamics of Solar-Grade-silicon refining Intermetallics,2003(11):1111-111715Peter Woditsch,Wolfgang Koch. Solar grade silicon feedstock supply fo PV ludustrv. Solar Energy Materials&Solar Cells,72(2002):11-7 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?1ar