分離爪零件的機械加工工藝規(guī)程及鉆φ3孔夾具設計

分離爪零件的機械加工工藝規(guī)程及鉆3孔夾具設計,分離爪零件的機械加工工藝規(guī)程及鉆3孔夾具設計,分離,零件,機械,加工,工藝,規(guī)程,夾具,設計 目錄一、 零件的分析1（一） 零件的作用1（二） 零件的工藝分析1（三） 零件圖1二、 工藝規(guī)程設計2(一)確定毛坯的制造形式2（二）基面的選擇3（三）制定工藝路線3（四）機械加工余量、工序尺寸及毛皮尺寸的確定4三、 夾具設計9（一）問題的提出10（二）確定結(jié)構(gòu)方案10(三)平衡配重設計11（四）設計制定夾具主要技術條件11（五）設計感想及體會11參考文獻13零件的分析（一） 零件的作用題目所給的零件是分離爪。它位于車床變速機構(gòu)中，主要起換檔的作用，使主軸回轉(zhuǎn)運動按照工作者的要求工作，工作過程：撥叉零件是在傳動系統(tǒng)中撥動滑移齒輪，以實現(xiàn)系統(tǒng)調(diào)速，轉(zhuǎn)向。（二） 零件的工藝分析C6132D車床共有兩處加工表面和一個空內(nèi)表面，其間有一定位置要求。分述如下：1， 零件孔40mm的上加工表面這一組加工表面包括：孔40mm的下加工表面，孔40mm的外表面，有粗糙度要求為Ra等于5um。加工時以50mm下端面為基準面，并加緊50的左右兩端面。2， 孔50mm下端面及孔36的內(nèi)表面其中50mm的下端面有粗糙度要求為Ra等于5um，這需要粗車；孔36mm的內(nèi)表面粗糙度要求5，需要進一步精車，這都以40空及端面定位，這就是先選一粗基準，然后必須要用已加工表面作為精基準根據(jù)各加工方法的經(jīng)濟精度及一般機床所能達到的位置精 度，該零件沒有很難加工的表面尺寸，上述表面的技術要求采用常規(guī)加工工藝均可以保證，對于這兩組加工表面而言，可以先加工其中一組表面，然后借助于專用夾具加工另一組表面，并且保證它們的位置精度要求（三） 零件圖二、 工藝規(guī)程設計(一)確定毛坯的制造形式1：零件材料為QT40-10。考慮零件在機床運行過程中所受沖擊不大，零件結(jié)構(gòu)又比較簡單，故選擇鑄件毛坯。2：已知零件的生產(chǎn)為大批生產(chǎn)，初步確定工藝安排的基本傾向為：加工過程劃分階段，工序都要集中，加工設備主要以專用設備為主，采用專用夾具，這樣生產(chǎn)質(zhì)量高，投產(chǎn)快生產(chǎn)率較高，能適應大批量需求3：根據(jù)零件的材料確定毛坯為鑄件，材料為QT40-10，毛坯的鑄造方法為砂型型鑄造，鑄件尺寸公差等級為IT13級以下。（二）基面的選擇基面選擇是工藝規(guī)程設計中的重要工作之一?；孢x擇得正確與合理可以使加工質(zhì)量得到保證，生產(chǎn)率得以提高。（1）粗基準的選擇。對于零件而言，盡可能選擇不加工表面為粗基準。而對有若干個不加工表面的工件，則應以與加工表面要求相對位置精度較高的不加工表面作粗基準。根據(jù)這個基準選擇原則，現(xiàn)選取50孔的上表面作為粗基準，利用圓柱來定位，另外四爪握住兩邊來限制六個自由度，達到完全定位,然后進行車削。（2）精基準的選擇。主要應該考慮基準重合的問題。（三）制定工藝路線制定工藝路線，在生產(chǎn)綱領確定的情況下, 根據(jù)零件的幾何形狀、尺寸精度及位置精度等技術要求來制定工藝路線。可以考慮采用萬能性機床配以專用工卡具,并盡量使工序集中來提高生產(chǎn)率。除此之外，還應當考慮經(jīng)濟效果，以便使生產(chǎn)成本盡量下降。1.工藝路線方案工序一 鑄造。工序二 低溫退火熱處理。工序三 以50mm的孔及下端面定位，車40孔，并車孔的上端面。工序四 以40內(nèi)孔及上端面定位，車50下端面及36mm孔。工序五 以40內(nèi)孔及上端面定位，精車36及36.5孔。工序六 以36內(nèi)孔及端面定位，精車40外圓倒角。 工序七 以36內(nèi)孔及端面定位，銑8.3mm孔兩端至尺寸8+0.5mm，保證30mm尺寸至300.03mm，表面粗糙度要求為5mm。工序八 以36內(nèi)孔及端面定位，鉆、鉸8.3mm至尺寸。工序九 以36內(nèi)孔及端面定位，鉆3的孔，孔口倒角。工序十 以36內(nèi)孔及端面定位，精銑螺旋面工序十一 去毛刺。工序十二 檢驗。2，選擇機床（1） 工序三，工序四，工序五，工序六均為車表面，可用C6132D車床。（2） 工序七為銑削工藝，可采用銑床XD6132。() 工序八、九為鉆，鉸工藝，用臺式鉆床Z402。() 工學十需精銑螺旋面，很復雜，選用數(shù)控立式升降臺銑床XK50253，選擇夾具:每個工序都可以采用專用的夾具,夾具后面再設計。4，選擇刀具:在銑床上用硬質(zhì)合金端銑刀，銑上下兩槽用三面刃銑刀，在車床上選硬質(zhì)合金外圓車刀和端面車刀，鉆床用直柄麻花鉆和錐柄機用鉸刀，數(shù)控銑床用專用螺旋面銑刀5，選擇量具:精度要求較高的可用內(nèi)徑千分尺量程50125，其余用游標卡尺分度值為0.02（四）機械加工余量、工序尺寸及毛皮尺寸的確定”分離爪”；零件材料為QT40-10，生產(chǎn)類型大批量，鑄造毛坯。據(jù)以上原始資料及加工路線，分別確定各加工表面的機械加工余量、工序尺寸及毛坯尺寸如下：1. 查機械制造工藝設計簡明手冊（以下稱工藝手冊銑削加工余量為：粗銑 2-4mm半精銑 1-2mm精銑 0-1mm2圓柱（40用鑄成）內(nèi)孔8.3用鉆和鉸成，加工余量：鉆孔 1mm3粗車加工余量： 粗 車 1.2mm 半精車 0.8mm由于本設計規(guī)定的零件為大批量生產(chǎn)，應該采用調(diào)整加工。因此在計算最大、最小加工余量時應按調(diào)整法加工方式予以確認。（五）確立切削用量及基本工時工序一 毛坯工序 用鑄造尺寸如毛坯圖（CAD毛坯圖）工序二以36mm孔及端面定位銑8.3孔上下兩端面1. 加工條件工件材料：QT40-10，b=16GPa，HB=190241,鑄造。加工要求：粗銑8.3上下兩端面面。機床：銑床，XD6132臥式銑床。刀具：硬質(zhì)合金鋼端銑刀,牌號YG6。銑削寬度ae=40mm,深度apPcc。故校驗合格。最終確定 ap=1.3mm，nc=475r/min,Vfc=475mm/s，V c=119.3m/min，f z=0.16mm/z。6）計算基本工時tmL/ Vf=(40+40)/475=0.168min。工序三 以36及端面為精基準，鉆、鉸8.3孔 1. 選擇鉆頭 選擇直柄麻花鉆鉆頭，粗鉆時do=8mm，鉆頭采用雙頭刃磨法，后角o12，二重刃長度=2.5mm,橫刀長b=1.5mm,寬l=3mm,棱帶長度 2.選擇切削用量 （1）決定進給量查切 ，所以， 按鉆頭強度選擇 按機床強度選擇最終決定選擇機床已有的進給量 經(jīng)校驗 校驗成功。 （2）鉆頭磨鈍標準及壽命后刀面最大磨損限度（查切）為0.50.8mm，壽命（3）切削速度查切 修正系數(shù) 故。查切機床實際轉(zhuǎn)速為故實際的切削速度（4）校驗扭矩功率 所以 故滿足條件，校驗成立。3.計算工時T=L/nf=65/272*1.45=0.163min 工序四 以36及端面為精基準，鉆、鉸3mm孔，倒角 1. 選擇鉆頭 選擇粗直柄麻花鉆鉆頭，粗鉆時do=3mm，鉆頭采用雙頭刃磨法，后角o4，二重刃長度=1.5mm,橫刀長b=1mm,寬l=3mm,棱帶長度 2.選擇切削用量 （1）決定進給量查切 f=0.3-0.5mm/r ，所以， 按鉆頭強度選擇 按機床強度選擇最終決定選擇機床已有的進給量 經(jīng)校驗 校驗成功。 （2）鉆頭磨鈍標準及壽命后刀面最大磨損限度（查切）為0.50.8mm，壽命（3）切削速度查切 修正系數(shù) 故。查切機床實際轉(zhuǎn)速為故實際的切削速度（4）校驗扭矩功率 所以 故滿足條件，校驗成立。3.計算工時T=L/nf=65/272*1.45=0.163min 工序十一 去毛刺工序十二 檢查。其余幾步數(shù)據(jù)見工藝卡片。三、 夾具設計經(jīng)過仔細分析決定設計鉆床夾具。在確定夾具設計方案時應當遵循的原則是： 確保工件的加工質(zhì)量； 工藝性好，結(jié)構(gòu)盡量簡單； 使用性好，操作省力高效； 定位、夾緊快速、準確，能提高生產(chǎn)率； 經(jīng)濟性好，制造成本低廉。確定最合理的設計方案。 確定定位方案，設計定位裝置。定位應符合“六點定位原則”。定位元件盡可能選用標準件。 確定夾緊方案，設計夾緊機構(gòu)。夾緊可以用手動、氣動、液壓或其它動力源。重點應考慮夾緊力的大小、方向、作用點，以及作用力的傳遞方式，保證不破壞定位，不造成工件過量變形，不會有活動度為零的“機構(gòu)”，并且應滿足生產(chǎn)率的要求。 確定夾具整體結(jié)構(gòu)方案。定位、夾緊確定之后，還要確定其它機構(gòu)，如對刀裝置、導引元件、分度機構(gòu)、頂出裝置等。最后設計夾具體，將各種元件、機構(gòu)有機地連接在一起。 夾具精度分析。在繪制的夾具結(jié)構(gòu)草圖上，標注出初步確定的定位元件的公差配合關系及相互位置精度，然后計算定位誤差，根據(jù)誤差不等式關系檢驗所規(guī)定的精度是否滿足本工序加工技術要求，是否合理。否則應采取措施（如重新確定公差、更換定位元件、改變定位基準，必要時甚至改變原設計方案），然后重新分析計算。 夾具夾緊力分析。首先應計算切削力大小，它是計算夾緊力的主要依據(jù)。通常確定切削力有以下三種方法： 由經(jīng)驗公式算出； 由單位切削力算出； 由手冊上提供的諾模圖（如M-P-N圖）查出。根據(jù)切削力、夾緊力的方向、大小，按靜力平衡條件求得理論夾緊力。為了保證工件裝夾的安全可靠，夾緊機構(gòu)（或元件）產(chǎn)生的實際夾緊力，一般應為理論夾緊力的倍。（一）問題的提出在給定的零件中，對本步加工的定位提出具體的要求是一定的公差，定位要求較高。因此，本步的重點應在保證工件的定位，來保證兩孔的尺寸。具體要求：1、保證孔1，孔2的尺寸。 2、保證圖中其它的標尺寸。（二）確定結(jié)構(gòu)方案方案一：兩小頭孔外圓用一邊為固定型塊，另一邊用可松緊的活動型塊來夾緊工件。此結(jié)構(gòu)的優(yōu)點在于結(jié)構(gòu)十分簡單，并且易于實際應用。但其同時有致命的缺點夾緊精度不能保證，并且豎直方向沒有任何夾緊力，這使得零件加工出來的精度質(zhì)量并不高，并且在進行加工時會發(fā)生翹曲形變。方案二：根據(jù)得到題目要求，需要加工的工序是鉆直徑為的小頭孔，因此工件左端用活動型塊固定小頭孔外圓，另使用定位銷在中心大孔處進行定位。上端使用開口墊圈進行固定，這樣在保證豎直方向夾緊力的同時，又方便加工零件的拆裝。在夾具底座上鑄造有鉆模板，雖然相對來說結(jié)構(gòu)略微復雜，但是真正實際生產(chǎn)當中的操作并不繁瑣，甚至要較方案更加簡便。最后確定選擇方案一。(三)平衡配重設計平衡配重的計算通常較復雜，對于設計來說，只在夾具體上設計了配重塊，配重的調(diào)整需根據(jù)實際情況進行。 （四）設計制定夾具主要技術條件 1： 首先保證孔1，孔2的尺寸和表面粗糙度不大于。 2：其次要保證外圓的尺寸。 3：同時保證外圓端面粗糙度。 4：繪制裝配圖及零件圖（附）。（五）設計感想及體會為期三周的夾具課程設計已經(jīng)接近尾聲，回顧整個過程，通過老師和同學的幫助，還有自己不懈的努力，終于定時定量的完成了這次課程設計。課程設計作為機械制造技術基礎課程的重要環(huán)節(jié)，使理論與實踐更加接近，加深了理論知識的理解，強化了生產(chǎn)實習中的感性認識。本次課程設計主要經(jīng)歷了兩個階段：第一段是接卸加工工藝規(guī)程設計，第二階段是專用夾具設計。第一階段運用了基準選擇、切削用量選擇計算、機床選用、實踐定額計算等方面的知識；夾具設計的階段運用了工件定位、夾緊機構(gòu)及零件結(jié)構(gòu)設計設計等方面的知識。通過此次設計，是我基本掌握了零件的加工過程分析、工藝文件的編制、專用夾具設計的方法和步驟等。學會了查相關手冊、選擇使用工藝裝備等等。本設計還存在很多不足之處。一是由于本人的畫圖不夠純熟，在畫圖遇到復雜的和難的地方時顯力不從心，使圖沒達到預期設計出來的水平。二是對分離爪的工作原理掌握的不夠熟練，在設計過程中不能全面地考慮問題，造成走許多彎路，設計速度緩慢，這些都需要進一步研究和進一步實踐來解決?？偟膩碚f，這次設計，使我們在基本理論的綜合運用及正確解決實際問題等方面得到了一次較好的訓練。提高了我們的思考、解決問題創(chuàng)新設計的能力，為以后的設計工作打下了較好的基礎。 由于能力所限，設計還有許多不足之粗，懇請老師們批評指正。 參考文獻1. 機械工業(yè)出版社.機械制造技術基礎課程設計指導教程2. 機械工業(yè)出版社. 互換性與測量技術基礎第3版3. 高等教育出版社. 機械制圖4. 機械工業(yè)出版社. 機械制造技術基礎第2版5.主編：吳兆祥，機械制造技術課程設計，浙江大學出版社6.主編：蔣建強，數(shù)控加工技術與實訓，北京：電子工業(yè)出版社7.主編：李啟炎，計算機繪圖（初級）AUTOCAD2004版，同濟大學出版社8.主編：夏鳳芳，數(shù)控機床，高等教育出版社9.主編：廖兆榮，數(shù)控幾雙電氣控制，高等教育出版社，10.機械工程手冊工程材料，1996年第二版11.主編：成大先，機械設計手冊北京：機械工業(yè)出版社12.主編：甘永立，幾何量公差與檢測上海：上?？茖W技術出版社材料成型及控制工程機械加工工序卡片工序名稱車削工序號10零件名稱 分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10150-180球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號臥式車床C6132D專用車夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)切 削 深 度進給量主 軸轉(zhuǎn) 速切 削速 度基 本工 時12粗車40外圓至尺寸42mm 粗車端面保證尺寸16mm硬質(zhì)合金外圓車刀硬質(zhì)合金端面車刀游標卡尺 35mm1 2mm0.21mm/z0.21mm/z300r/min 44.27mm/min4.6min設 計 者指 導 教 師共 頁第 1 頁材料成型及控制工程機械加工工序卡片工序名稱 車削工序號20零件名稱 分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10170220球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號臥式車床C6132D專用車夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)切 削 深 度進給量主 軸轉(zhuǎn) 速切 削速 度基 本工 時12粗車大端面 粗車36的孔硬質(zhì)合金外圓車刀硬質(zhì)合金端面車刀游標卡尺36mm12.5mm0.2mm/z300r/min33.91mm/min4.3min設 計 者指 導 教 師共 頁第 2 頁材料成型及控制工程機械加工工序卡片工序名稱車削工序號30零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10150-180球墨鑄鐵設 備夾 具名 稱輔 助工 具名 稱型 號臥式車床C6132D專用車夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)切 削 深 度進給量主 軸轉(zhuǎn) 速切 削速 度基 本工 時12精車36, 36.5至尺寸要求硬質(zhì)合金外圓車刀游標卡尺35mm20.25mm0.10mm/z150r/min 16.96mm/min 17.72mm/min5.8min設 計 者指 導 教 師共 頁第 3 頁材料成型及控制工程機械加工工序卡片工序名稱車削工序號40零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10150-180球墨鑄鐵設 備夾 具名 稱輔 助工 具名 稱型 號臥式車床C6132D專用車夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)切 削 深 度進給量主 軸轉(zhuǎn) 速切 削速 度基 本工 時1精車40外圓至尺寸要求，倒角145度硬質(zhì)合金外圓車刀游標卡尺內(nèi)徑千分尺35mm20.25mm0.10mm/z150r/min18.84mm/min2.75min設 計 者指 導 教 師共 頁第 4 頁材料成型及控制工程機械加工工序卡片工序名稱銑削工序號50零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10170220球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號立式銑床X715專用銑夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)銑 削 深 度進給量主 軸轉(zhuǎn) 速銑 削速 度基 本工 時11銑8.3孔左端面，保證30mm尺寸至300.03，表面粗糙度要求為5um硬質(zhì)合金端銑刀游標卡尺8.3mm12.5mm0.18mm/r530r/min50m/min84s設 計 者指 導 教 師共 頁第 5 頁材料成型及控制工程機械加工工序卡片工序名稱銑削工序號零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10170220球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號立式銑床X715專用銑夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)銑 削 深 度進給量主 軸轉(zhuǎn) 速銑 削速 度基 本工 時11銑8.3孔右端面，表面粗糙度要求為5um硬質(zhì)合金端銑刀游標卡尺8.3mm12.5mm0.18mm/r530r/min50m/min84s設 計 者指 導 教 師共 頁第 6 頁材料成型及控制工程機械加工工序卡片工序名稱鉆工序號零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10170220球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號立式鉆床Z402專用鉆夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)銑 削 深 度進給量主 軸轉(zhuǎn) 速銑 削速 度基 本工 時1鉆，鉸8.3的孔至要求尺寸直柄麻花鉆錐柄機用鉸刀內(nèi)徑千分尺18mm10.4mm/z0.mm/z0.m/z400r/min630r/min630r/min29.7m/min15 m/min29.7 m/min25s25s 25s設 計 者指 導 教 師共 頁第 7 頁材料成型及控制工程機械加工工序卡片工序名稱鉆工序號零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10170220球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號立式鉆床Z402專用鉆夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)銑 削 深 度進給量主 軸轉(zhuǎn) 速銑 削速 度基 本工 時1鉆3mm孔，孔口倒角145度直柄麻花鉆內(nèi)徑千分尺15mm1480r/min520r/min58.34m/min37.25m/min61s25s設 計 者指 導 教 師共 頁第 頁材料成型及控制工程機械加工工序卡片工序名稱銑削工序號零件名稱分離爪零件號1零件重量同時加工零件數(shù)1材 料毛 坯牌 號硬 度型 號重 量QT40-10170220球墨鑄鐵0.72kg設 備夾 具名 稱輔 助工 具名 稱型 號數(shù)控立式升降臺銑床XK5025專用銑夾具安 裝工 步安裝及工步說明刀 具量 具走 刀長 度走 刀次 數(shù)銑 削 深 度進給量主 軸轉(zhuǎn) 速銑 削速 度基 本工 時11精銑螺旋面173s設 計 者指 導 教 師共 頁第 頁材料成型及控制工程機 械 加 工 工 藝 過 程 卡 片零件號零 件 名 稱1分離爪工序號工 序 名 稱設 備夾 具刀 具量 具名 稱型 號名 稱 名 稱名 稱102030405060708090100110120130砂型鑄造熱處理粗車40外圓至尺寸42mm粗車端面保證尺寸16mm粗車大端面粗車36的孔精車36, 36.5至尺寸要求精車40外圓至尺寸要求，倒角145度銑8.3孔左端面，保證30mm尺寸至300.03，表面粗糙度要求為5um銑8.3孔右端面，表面粗糙度要求為5um鉆，鉸8.3的孔至要求尺寸鉆3mm孔，孔口倒角145度精銑螺旋面去毛刺檢驗臥式車床臥式車床臥式車床臥式車床立式銑床立式銑床立式鉆床立式鉆床立式銑床C6132DC6132DC6132DC6132DX715X715Z402Z402XK5025專用車夾具專用車夾具專用車夾具專用車夾具專用銑夾具專用銑夾具專用鉆夾具專用鉆夾具專用銑夾具硬質(zhì)合金外圓車刀硬質(zhì)合金端面車刀硬質(zhì)合金外圓車刀硬質(zhì)合金端面車刀硬質(zhì)合金外圓車刀硬質(zhì)合金外圓車刀硬質(zhì)合金端銑刀硬質(zhì)合金端銑刀直柄麻花鉆錐柄機用鉸刀直柄麻花鉆螺旋銑刀游標卡尺游標卡尺游標卡尺游標卡尺游標卡尺游標卡尺游標卡尺游標卡尺內(nèi)徑千分尺內(nèi)徑千分尺游標卡尺Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained constraint status, a workpiece under any locating scheme falls into one of the following three categories: locating problem using screw theory in 1989. It is concluded that the locating wrenches matrix needs to be full rank to achieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. 3 considered ARTICLE IN PRESS 0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2004.11.012 C3 Corresponding author. Tel.: +15088316092; fax: +15088316412. E-mail address: hsongwpi.edu (H. Song). 1. Well-constrained (deterministic): The workpiece ismatedat auniqueposition when six locatorsare madeto contact the 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 the basis of analytical investigations for deterministic locating that followed. Chou et al. 2 formulated the deterministic 1. Introduction A xture is a mechanism used in manufacturing operations to hold a workpiece rmly in position. Being a crucial step in process planning for machining parts, xture design needs to ensure the positional accuracy and dimensional accuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a location scheme. V-block and pin-hole locating principles are also commonly used. Alocationschemeforamachiningxturemustsatisfyanumberofrequirements.Themostbasicrequirementisthat it must provide deterministic location for the workpiece 1. This notion states that a locator scheme produces deterministic location when the workpiece cannot move without losing contact with at least one locator. This has been one of the most fundamental guidelines for xture design and studied by many researchers. Concerning geometry Abstract Geometry constraint is one of the most important considerations in xture design. Analytical formulation of deterministic location has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process of actual xture design practice has not been thoroughly studied. In this paper, a methodology to characterize xturing systems geometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognized withgiven locatingscheme.All un-constrainedmotionsofaworkpiece inanunder-constraintstatuscanbeautomaticallyidentied. This assists the designer to improve decit locating scheme and provides guidelines for revision to eventually achieve deterministic locating. r 2005 Elsevier Ltd. All rights reserved. CAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA Received 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004 Locating completeness evaluation and revision in xture plan H. Song C3 , Y. Rong locatorworkpiece contact area effects instead of applying point contact. They introduced a contact matrix and pointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson 4 suggested that a linear approximation may not be sufcient for some applications such as non-prismatic surfaces or non-small relative errors.Heproposed asecond-order Taylor expansionwhichalsotakes locatorerror interaction into account. Marin and Ferreira 5 applied Chous formulation on 3-2-1 location and formulated several easy-to-follow planning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyze non-deterministic location. In the Asada and Bys formulation, they assumed frictionless and point contact between xturing elements and workpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surface function is g i (as shown in Fig. 1). The surface function isdened as g i q C3 0: To be deterministic, there should be a unique solution for the following equation set for all locators. g i q0; i 1;2; .; n, (1) where n is the number of locators and q x 0 ; y 0 ; z 0 ;y 0 ;f 0 ;c 0 C138 represents the position and orientation of the workpiece. Only considering the vicinity of desired location q C3 ; where q q C3 Dq; Asada and By showed that ARTICLE IN PRESS H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 369 g i qg i q C3 h i Dq, (2) where h i is the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locating requirement can be satised if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solution q q C3 : rank qg 1 qx 0 qg 1 qy 0 qg 1 qz 0 qg 1 qy 0 qg 1 qf 0 qg 1 qc 0 : qg i qx 0 qg i qy 0 qg i qz 0 qg i qy 0 qg i qf 0 qg i qc 0 : qg n qx 0 qg n qy 0 qg n qz 0 qg n qy 0 qg n qf 0 qg n qc 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 8 : 9 = ; 6. (3) Upongivena3-2-1locatingscheme, therankofaJacobianmatrixforconstraintequationstellstheconstraintstatus as shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one free motion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme has more than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that it can 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 6 has developed a systematic approach for this purpose. Hence, this criterion can be applied to all locating schemes. X Y Z O X Y Z O (x 0 ,y 0 ,z 0 ) g i UCS WCS Workpiece Fig. 1. Fixturing system model. They further introduced several indexes derived from those matrixes to evaluate locator congurations, followed by optimization through constrained nonlinear programming. Their analytical study, however, does not concern the ARTICLE IN PRESS revision of non-deterministic locating. Currently, there is no systematic study on how to deal with a xture design that failed to provide deterministic location. 2. Locatingcompletenessevaluation If deterministic location is not achieved by designed xturing system, it is as important for designers to know what the constraint status is and how to improve the design. If the xturing 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 locating scheme more efciently. A general strategy to characterize geometry constraint status of a locating scheme is described in Fig. 2. In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status (see Appendix for derivation of locating matrix). The deterministic locating requires six locators that provide full rank locating matrix W L : As shown in Fig. 3, for given locator number n; locating normal vector a i ; b i ; c i C138 and locating position x i ; y i ; z i C138 for each locator, i 1;2; .; n; the n C26 locating matrix can be determined as follows: a 1 b 1 c 1 c 1 y 1 C0 b 1 z 1 a 1 z 1 C0 c 1 x 1 b 1 x 1 C0 a 1 y 1 : : : : 2 6 3 7 Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module in their automated computer-aided xture design verication system. Their CAFDV system can calculate the Jacobian matrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivity to locating error. Xiong et al. 8 presented an approach to check the rank of locating matrix W L (see Appendix). They also intro- duced left/right generalized inverse of the locating matrix to analyze the geometric errors of workpiece. It has been shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators are related as follows: Well-constrained : DX W L Dr, (4) Over-constrained : DX W T L W L C01 W T L Dr, (5) Under-constrained : DX W T L W L W T L C01 Dr I 6C26 C0 W T L W L W T L C01 W L l, (6) where l is an arbitrary vector. Table 1 Rank Number of locators Status o 6 Under-constrained 6 6 Well-constrained 6 46 Over-constrained H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370 W L a i b i c i c i y i C0 b i z i a i z i C0 c i x i b i x i C0 a i y i : : : : a n b n c n c n y n C0 b n z n a n z n C0 c n x n b n x n C0 a n y n 6 6 6 6 6 4 7 7 7 7 7 5 .(7) When rankW L 6 and n 6; the workpiece is well-constrained. When rankW L 6 and n46; the workpiece is over-constrained. This means there are n C06 unnecessary locators in the locating scheme. The workpiece will be well-constrained without the presence of those n C06 locators. The mathematical representationforthisstatusisthat thereare n C06 rowvectorsinlocating matrix thatcanbeexpressed as linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist one ARTICLE IN PRESS locat determ 1. 2. 3. 4. be 3. workpi H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 371 ing scheme that provides deterministic location. The developed algorithm uses the following approach to ine the unnecessary locators: Find all the combination of n C06 locators. For each combination, remove that n C06 locators from locating scheme. Recalculate the rank of locating matrix for the left six locators. If the rank remains unchanged, the removed n C06 locators are responsible for over-constrained status. This method may yield multi-solutions and require designer to determine which set of unnecessary locators should removed for the best locating performance. When rankW L o6; the workpiece is under-constrained. Algorithmdevelopmentandimplementation The algorithm to be developed here will dedicate to provide information on un-constrained motions of the ece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ Fig. 2. Geometry constraint status characterization. X Z Y (a 1 ,b 1 ,c 1 ) 2 ,b 2 ,c 2 ) (x 1 ,y 1 ,z 1 ) (x 2 ,y 2 ,z 2 ) (a i ,b i ,c i ) (x i ,y i ,z i ) (a Fig. 3. A simplied locating scheme. orient ij L L L ARTICLE IN PRESS 372 5. To identify allthe un-constrained motions oftheworkpiece, V dx i ;dy i ;dz i ;da xi ;da yi ;da zi C138 isintroducedsuchthat V DX 0. (9) Since rankDXo6; there must exist non-zero V that satises 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;0C138 says that the rotation about x-axisisnotconstrained. 0;1;1;0;0;0C138 meansthat theworkpiececanmovealongthedirection given by vector 0;1;1C138: There could be innite solutions. The solution space, however, can be constructed by 6C0 rankW L basic solutions. Following analysis is dedicated to nd out the basic solutions. From Eqs. (8) and (9) VX dxDx dyDy dzDz da x Da x da y Da y da z Da z dx X n i1 w 1i Dr i dy X n i1 w 2i Dr i dz X n i1 w 3i Dr i da x X n i1 w 4i Dr i da y X n i1 w 5i Dr i da z X n i1 w 6i Dr i X n i1 Vw 1i ; w 2i ; w 3i ; w 4i ; w 5i ; w 6i C138 T Dr i 0. 10 Eq. (10) holds for 8Dr i if and only if Eq. (11) is true for 8i1pipn: Vw 1i ; w 2i ; w 3i ; w 4i ; w 5i ; w 6i C138 T 0. (11) Eq. (11) illustrates the dependency relationships among row vectors of W r : In special cases, say, all w 1j equal 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 to understand because Dx 0 in this case, implying that the corresponding position error of the workpiece is not dependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combined motion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. For instance, 9w 1j a0;w 2j a0; w 1j C0w 2j for 8j: Inthisscenario,theworkpiece cannotmovealong x-ory-axis.However,it can move along the diagonal line between x-andy-axis dened by vector 1, 1, 0. To nd solutions for general cases, the following strategy was developed: 1. Eliminate dependent row(s) from locating matrix. Let r rank W L ; n number of locator. If ron; create a vector in n C0 r dimension space U u 1 : u j : u nC0r hi 1pjpn C0 r; 1pu j pn: Select u j in the way that rankW L r still holds after setting all the terms of all the u j th row(s) equal to zero. Set r C26 modied locating matrix W LM a 1 b 1 c 1 c 1 y 1 C0 b 1 z 1 a 1 z 1 C0 c 1 x 1 b 1 x 1 C0 a 1 y 1 : : : : a i b i c i c i y i C0 b i z i a i z i C0 c i x i b i x i C0 a i y i : : : : a n b n c n c n y n C0 b n z n a n z n C0 c n x n b n x n C0 a n y n 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 rC26 , wher geomet ation errors and locator errors can be expressed as follows: DX Dx Dy Dz a x a y a z 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 w 11 : w 1i : w 1n w 21 : w 2i : w 2n w 31 : w 3i : w 3n w 41 : w 4i : w 4n w 51 : w 5i : w 5n w 61 : w 6i : w 6n 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 C1 Dr 1 : Dr i : Dr n 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 , (8) e Dx;Dy;Dz;a x ;a y ;a z are displacement along x, y, z axis and rotation about x, y, z axis, respectively. Dr i is ric error of the ith locator. w is dened by right generalized inverse of the locating matrix W r W T W W T C01 H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 where i 1;2; :; niau j : 4. 6. constr Exampl vector ARTICLE IN PRESS L 3 : 0, 0, 1 0 , 2, 1, 0 0 , L 4 : 0, 1, 0 0 , 3, 0, 2 0 , L 5 : 0, 1, 0 0 , 1, 0, 2 0 . Consequently, the locating matrix is determined. W L 001 3 C010 001 3 C030 001 1 C020 010C0203 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . L L v s : v 6 6 6 6 4 7 7 7 5 w q k i : w q k r 6 6 6 4 7 7 7 5 C1 w l1 : w li : w lr : w 61 : w 6i : w 6r 6 6 6 4 7 7 7 5 , where s 1;2; :;6saq j ; saq k ; l 1;2; :;6 laq j : Repeat step 4 (select another term from Q) and step 5 until all 6C0 r basic solutions have been determined. Based on this algorithm, a C+ program was developed to identify the under-constrained status and un- ained motions. e1. In a surface grinding operation, a workpiece is located on a xture system as shown in Fig. 4. The normal and position of each locator are as follows: 1 : 0, 0, 1 0 , 1, 3, 0 0 , 2 : 0, 0, 1 0 ,3,3,0 0 , Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found as follows. v 1 : 2 6 6 6 3 7 7 7 w q k 1 : 2 6 6 6 3 7 7 7 w 11 : w 1i : w 1r : 2 6 6 6 3 7 7 7 C01 5. W rm w l1 : w li : w lr : w 61 : w 6i : w 6r 6 6 6 4 7 7 7 5 6C26 , where l 1;2; :;6 laq j : Normalize the free motion space. Suppose V V 1 ; V 2 ; V 3 ; V 4 ; V 5 ; V 6 C138 is one of the basic solutions of Eq. (10) with all six terms undetermined. Select a term q k from vector Q1pkp6C0 r: Set V q k C01; V q j 0 j 1;2; :;6C0 r; jak; ( 2. Compute the 6C2 n right generalized inverse of the modied locating matrix W r W T LM W LM W T LM C01 w 11 : w 1i : w 1r w 21 : w 2i : w 2r w 31 : w 3i : w 3r w 41 : w 4i : w 4r w 51 : w 5i : w 5r w 61 : w 6i : w 6r 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 6C2r 3. Trim W r down to a r C2 rfull rank matrix W rm : r rankW L o6: Construct a 6C0 r dimension vector Q q 1 : q j : q 6C0r hi 1pjp6C0 r; 1pq j pn: Select q j in the way that rankW r r still holds after setting all the terms of all the q j th row(s) equal to zero. Set r C2 r modied inverse matrix w 11 : w 1i : w 1r : 2 6 6 6 3 7 7 7 H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 373 010C0201 ARTICLE IN PRESS This locating system provides under-constrained positioning since rankW L 5o6: The program then calculates the right generalized inverse of the locating matrix. W r 00 000 0:50:5 C01 C00:51:5 0:75 C01:25 1:50 0 0:25 0:25 C00:50 0 0:5 C00:5000 0000:5 C00:5 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The rst row is recognized as a dependent row because removal of this row does not affect rank of the matrix. The other ve rows are independent rows. A linear combination of the independent rows is found according the requirementinstep5oftheprocedureforunder-constrainedstatus.Thesolutionforthisspecialcaseisobviousthatall the coefcients are zero. Hence, the un-constrained motion of workpiece can be determined as V C01; 0; 0; 0; 0; 0C138: This indicates that the workpiece can move along x direction. Based on this result, an additional locator should be employed to constraint displacement of workpiece along x-axis. X Z Y L 3 L 4 L 5 L 2 L 1 Fig. 4. Under-constrained locating scheme. H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374 Example2. Fig. 5 shows a knuckle with 3-2-1 locating system. The normal vector and position of each locator in this initial design are as follows: L 1 : 0, 1, 0 0 , 896, C0877, C0515 0 , L 2 : 0, 1, 0 0 , 1060, C0875, C0378 0 , L 3 : 0, 1, 0 0 , 1010, C0959, C0612 0 , L 4 : 0.9955, C00.0349, 0.088 0 , 977, C0902, C0624 0 , L 5 : 0.9955, C00.0349, 0.088 0 , 977, C0866, C0624 0 , L 6 : 0.088, 0.017, C00.996 0 , 1034, C0864, C0359 0 . The locating matrix of this conguration is W L 0 1 0 515:000:8960 01 0378: 1:0600 0 1 0 612:00:0100 0:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:8638 0:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:8280 0:0880 0:0170 C00:9960 866:6257998 :2466 0:0936 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 , rankW L 5o6 reveals that the workpiece is under-constrained. It is found that one of the rst ve rows can be removed without varying the rank of locating matrix. Suppose the rst row, i.e., locator L 1 is removed from W L ; the ARTICLE IN PRESS modied locating matrix turns into W LM 010378:001:0600 0 1 0 612: :0100 0:9955 C00:0349 0:0880 C0101:2445 C0707:2664 0:8638 0:9955 C00:0349 0:0880 C098:0728 C0707:2664 0:8280 0:0880 0:0170 C00:996 866:6257998 :2466 0:0936 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 . The right generalized inverse of the modied locating matrix is W r 1:8768 C01:8607 C020:6665 21:3716 0:4995 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . The program checked the dependent row and found every row is dependent on other ve rows. Without losing generality, the rst row is regarded as dependent row. The 5C25 modied inverse matrix is 2 3 Fig. 5. Knuckle 610 (modied from real design). H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378 375 W rm 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 6 6 6 6 6 6 4 7 7 7 7 7 7 5 . The undetermined solution is V C01; v 2 ; v 3 ; v 4 ; v 5 ; v 6 C138: To calculate the ve undetermined terms of V according to step 5, 1:8768 C01:8607 C020:6665 21:3716 0:4995 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 T C1 3:0551 C02:0551 C032:4448 32:4448 0 C01:0956 1:0862 12:0648 C012:4764 C00:2916 C00:0044 0:0044 0:0061 C00:0061 0 0:0025 C00:0025 0:0065 C00:0069 0:0007 C00:0004 0:0004 0:0284 C00:0284 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 C01 0; C01:713; C00:0432; C00:0706; 0:04C138. Substituting this result into the undetermined solution yields V C01;0; C01:713; C00:0432; C00:0706; 0:04C138 This vector represents a free motion dened by the combination of a displacement along C01, 0, C01.713 direction combine