傳動齒輪工藝夾具設(shè)計,傳動齒輪工藝夾具設(shè)計,傳動,齒輪,工藝,夾具,設(shè)計 傳動齒輪設(shè)計說明書 摘要： 隨著科技技術(shù)的不斷進步，生產(chǎn)都向著自動化、專業(yè)化和大批量化的方向發(fā)展。這就要求企業(yè)的生產(chǎn)在體現(xiàn)人性化的基礎(chǔ)上降低工人的生產(chǎn)強度和提高工人的生產(chǎn)效率，降低企業(yè)的生產(chǎn)成本?，F(xiàn)代的生產(chǎn)和應(yīng)用設(shè)備多數(shù)都采用機電一體化、數(shù)字控制技術(shù)和自動化的控制模式。在這種要求下齒輪零件越發(fā)體現(xiàn)出其廣闊的應(yīng)用領(lǐng)域和市場前景。特別是近年來與微電子、計算機技術(shù)相結(jié)合后，使齒輪零件進入了一個新的發(fā)展階段。在齒輪零部件是最重要部分，因需求的增加，所以生產(chǎn)也步入大批量化和自動化。 為適應(yīng)機械設(shè)備對齒輪加工的要求，對齒輪加工要求和技術(shù)領(lǐng)域的拓展還需要不斷的更新與改進。 關(guān)鍵詞：工藝設(shè)計 齒輪零件 齒輪傳動 abstract: Preface technology with the progress toward the production of automation, specialization and a large number of quantization direction. This requires the production of the human embodiment of the workers on the basis of reducing the intensity of production and enhance worker productivity. lower their production costs. Modern production and application of the majority of equipment used electromechanical integration, digital control technology and automation control mode. In such a request Gear institutions increasingly reflects its broad application areas and market prospects. Especially in recent years and microelectronics, computer technology integration, make technology gear drive has entered a new stage of development. Gear drive gear parts is the most important part, Gear by the relative movement of the drive to promote change in direction. Gear result of the design requirements are relatively strict, in order to adapt to the type of gear increasing and updating. Because of the increase in demand, production has entered a large number of quantitative and automation. To meet the mechanical equipment to gear machining requirements, Gear on the design requirements and technical fields is also expanding the need to constantly update and improve. Keywords:Gear Gear drive 目錄 1畢業(yè)設(shè)計工藝要求的基本任務(wù)和要求--------------------------------------1 1.1基本任務(wù)----------------------------------------------------------------------------------1 1.1.1工藝設(shè)計的基本任務(wù)---------------------------------------------------------------- 1 1.1.2夾具設(shè)計的基本任務(wù)-----------------------------------------------------------------2 1.2設(shè)計要求----------------------------------------------------------------------------------3 1.2.1工藝設(shè)計的設(shè)計要求-----------------------------------------------------------------3 1.2.2夾具設(shè)計的設(shè)計要求-----------------------------------------------------------------3 2畢業(yè)設(shè)計工藝設(shè)計的方法和步驟 -------- -------------------- ------- ------3 2.1生產(chǎn)綱領(lǐng)的計算與生產(chǎn)類型的確定-------------------------------------------------4 2.2分析零件圖-------------------------------------------------------------------------------4 2.3確定生產(chǎn)類型----------------------------------------------------------------------------4 2.4確定毛坯------------------------------------------------------------------------------- --5 2.5機械加工工藝過程--------------------------------------------------------------------- -5 2.6選擇機床和工藝設(shè)備---------------------------------------------------------------- ---6 2.7確定加工余量-------------------------------------------------------------------------- -7 2.8制作工藝卡片------------------------------------------------------------------------ ---7 3夾具設(shè)計--------------------------------------------------------------------------8 3.1夾具設(shè)計的目的和要求-------------------------------------------------------------- -8 3.2夾緊力的計算------------------------------------------------------------------------- -9 3.3夾具零件圖------------------------------------------------------------------------ -----10 致謝 參考文獻 1、畢業(yè)設(shè)計工藝要求的基本任務(wù)和要求 1.1、基本任務(wù) 1.1.1、工藝設(shè)計的基本任務(wù) (1)繪制零件工作圖一張 (2)繪制毛坯-零件合圖一張 (3)編制機械加工工藝規(guī)程卡片一套 (4)編寫設(shè)計說明書一份 1.1.2、夾具設(shè)計的基本任務(wù) (1)收集資料,為夾具設(shè)計做好準(zhǔn)備 (2)繪制草圖,進行必要的理論計算和分析以及夾具的結(jié)構(gòu)方案 (3)繪制總圖和主要非標(biāo)準(zhǔn)件零件圖,編寫設(shè)計說明書 (4)編制夾具的使用說明或技術(shù)要求 1.2、設(shè)計要求 1.2.1、工藝設(shè)計的設(shè)計要求 (1)保證零件加工質(zhì)量,達到圖紙的技術(shù)要求 (2)在保證加工質(zhì)量的前提下,盡可能提高生產(chǎn)效率 (3)要盡量減輕工人的勞動強度,生產(chǎn)安全 (4)在立足企業(yè)的前提下,盡可能采用國內(nèi)技術(shù)和裝備 (5)工藝規(guī)程應(yīng)正確.清晰,規(guī)范化,標(biāo)準(zhǔn)化的要求 1.2.2、夾具設(shè)計的設(shè)計要求 (1)保證工件的加工精度 (2)提高生產(chǎn)效率 (3)工藝性好 (4)使用性好 (5)經(jīng)濟性好 2、畢業(yè)設(shè)計工藝課程設(shè)計的方法和步驟 2.1、生產(chǎn)綱領(lǐng)的計算與生產(chǎn)類型的確定 生產(chǎn)綱領(lǐng)的大小對生產(chǎn)組織和零件加工工藝過程起著重要的作用.它決定了各工序所需專業(yè)化和自動化的程度以及所選用的工藝方法和工藝裝備. 零件生產(chǎn)綱領(lǐng)可按下式計算. N=Qn(1+a%)(1+b%) 式中：N-----零件的生產(chǎn)綱領(lǐng)（件/臺） Q-----產(chǎn)品的年產(chǎn)量（臺/年） n-----每臺產(chǎn)品中，該零件的數(shù)量（件/臺） a%----零件的備品率 b% ---零件的平均廢品率 2.2分析零件圖 1、零件的作用 傳動齒輪,，它是齒輪的一個主要一種，其功用是傳遞運動和運動方向，以適應(yīng)傳動機構(gòu)運動的需要。 2、零件的工藝分析 傳動齒輪零件如圖所示，該零件主要加工表面及技術(shù)要求分析如下。 （1）同軸孔φ35H7，φ49H7和同軸外圓φ92.55k7, φ66的同軸度、徑向圓跳動公差等級為8～9級，表面粗糙度為Ra≤1.6μm.。加工時最好在一次裝夾下將兩孔或兩外圓同時加工。 （2）與基準(zhǔn)孔有垂直度要求的端面，其端面圓跳動公差等級為7級，表面粗糙度為Ra≤3.2μm。工藝過程安排時應(yīng)注意保證其位置精度。 （3）距中心平面74.5mm的兩側(cè)面,表面粗糙度為Ra≤6.3μm。 （4）φ35孔的尺寸精度要保證，但孔軸線的同軸度共差等級為9級及兩孔公共軸線對基準(zhǔn)孔（A1-A2）位置度公差值為0.06μm，應(yīng)予以重視。 （5）各外圓位置度公差為，齒輪輪齒輪的跳動公差要保證。 （6）該零件選用材料為20CrMnTi，這種材料具有低碳合金鋼的特性，切削性能和工藝性均較好。 由各加工方法的經(jīng)濟精度及一般機床所能達到的精度可知，該零件沒有很難加工的表面，各表面的技術(shù)要求采用常規(guī)加工工藝均可達到。但是在加工過程中應(yīng)注意齒輪端面的加工。 3零件圖 4熱處理工序的安排 熱處理工序主要用來改善材料的性能及消除應(yīng)力。熱處理的方法．次數(shù)和在工藝路線中的位置，應(yīng)根據(jù)零件材料和熱處理的目的而定。 熱處理安排為 毛坯→粗加工（粗車）→半精加工（半精）→精加工（精車） ↑ （去應(yīng)力退火或者時效去應(yīng)力） 齒輪加工工藝過程: 毛坯的選擇---熱處理1---齒坯加工---齒面加工---齒端的加工---熱處理2---精基準(zhǔn)的修正 2.3確定生產(chǎn)類型 已知零件的年生產(chǎn)綱領(lǐng)為10000件，零件質(zhì)量約為3.6kg，查表1-1可知其生產(chǎn)類型為大批量生產(chǎn)，初步確定工藝安排的基本思路為：加工過程劃分階段；工序適當(dāng)集中；加工設(shè)備以通用設(shè)備為主；大量采用專用工裝。這樣安排，生產(chǎn)準(zhǔn)備工作投資較少，生產(chǎn)效率較高，且轉(zhuǎn)產(chǎn)容易。 2.4確定毛坯 1、確定毛坯種類 根據(jù)零件材料確定毛坯為鍛件。根據(jù)其桔構(gòu)形狀、尺寸大小、生產(chǎn)類型和材料性能，毛坯的鑄造方法選用砂型機器造型。 2、確定鍛件及形狀 根據(jù)表1-6取加工余量等級為MA-G級。根據(jù)表1-7確定各表面的鍛件機械加工余量（查表前必須先選擇定位基準(zhǔn)，以便確定基本尺寸）。 2.5機械加工工藝過程設(shè)計 1、選擇定位基準(zhǔn) （1）選擇粗基準(zhǔn) 選擇內(nèi)孔和端面定位元件.先加工外圓，按“基準(zhǔn)先行”的原則，采用外圓及端面為粗基準(zhǔn)先加工內(nèi)孔。 （2）選擇精基準(zhǔn) 為了保證圓跳動要求，各主要圓柱表面均互為基準(zhǔn)加工，并盡量遵守“基準(zhǔn)重合”的原則。其余表面加工采用“一孔一端面的定位方式，即以端面及φ35H7內(nèi)孔為精基準(zhǔn)。這樣，基準(zhǔn)統(tǒng)一，定位穩(wěn)定，夾具結(jié)構(gòu)及操作也較簡單。 但必須提高這個內(nèi)孔的精度，以保證定位精度。 2、擬定工藝過程 （1）選擇表面加工方法 查表1-19~表1-21，根據(jù)各表面加工要求和各種加工方法所能達到的經(jīng)濟精度，選技零件主要表面的加工方法與方案如下： φ35H7內(nèi)孔加工：粗車）—半精車—精車。 φ92.55外圓加工：粗車—半精車—精車 齒輪端面加工: 滾齒—磨齒—精基準(zhǔn)修正 (2) 選擇加工機床 外圓和端面的加工采用CA6140車床,包括內(nèi)孔的加工; 齒輪端面的粗加工采用滾齒機加工,精加工采用磨齒機加工; 內(nèi)孔的磨削加工采用普通的磨床加工; 鍵槽加工采用拉車加工; (3)加工刀具的選擇 高速鋼是含有較多的W 、Mo 、Gr 、V的的高合金工具鋼，與碳素工具鋼和合金工具鋼相比，高速鋼具有較高的熱穩(wěn)定性（500℃～650℃時仍能切削），故高速鋼刀具允許使用切削速度較高。 高速鋼還具有良好的綜合性能。其強度和韌性是現(xiàn)有刀具材料中最高的（其抗彎強度是強硬質(zhì)合金的2～3倍，韌性是硬質(zhì)合金的9～10倍），具有一定的硬度和耐磨性，切削性能能滿足一般加工要求，高速鋼刀具制造工藝簡單，刃磨易獲得鋒利的切削刃，能鍛造，熱處理變形小，所以適合加工本零件。 車床上用硬質(zhì)的合金刀; 滾齒機是用專用的齒輪滾刀; 磨齒機上用專用的齒輪磨刀; 拉床上用矩形拉刀; （4）確定工藝過程方案 1）擬定方案。各表面加工方法及粗、精基準(zhǔn)已基本確定，現(xiàn)按照“先粗后精”、“先主后次”、“基準(zhǔn)先行”等原則，初步擬定工藝過程方案. 如下表所示: 工序號 工序名稱 工序內(nèi)容 01 鍛 鍛造毛坯 02 熱處理 對毛坯進行正火處理 03 車 粗車Φ35內(nèi)孔，留磨削余量0.4mm 04 車 粗車Φ92.55外圓，端面 05 車 半精車外圓。留磨削余量0.4mm 06 車 齒端的加工倒圓，倒尖，倒棱 07 滾齒 輪齒齒形的滾齒加工 08 熱處理 齒面的滲碳淬火熱處理58~64HRC 09 插鍵槽 插8*3.3的鍵槽。保證和A0.035的平行度與0.05的對稱度 10 倒角 倒角1×45°，1.5×45° 11 磨 外圓至尺寸 12 磨 磨內(nèi)孔至Φ35mm公差要求0~+0.027 13 磨 齒面磨削 14 鉗 去毛刺 15 檢 檢驗 2.7確定各工序切削用量 在單件小批生產(chǎn)中，各工序的切削用量一般由操作工人根據(jù)具體情況自己確定，以簡化工藝文件。 在大批大量生產(chǎn)中則應(yīng)科學(xué)地．嚴(yán)格地選擇切削用量，以充分發(fā)揮高效率設(shè)備的潛力和作用。切削用量的選用與下列因素有關(guān)：生產(chǎn)率，加工質(zhì)量（主要是表面粗糙），切削力所收起的工藝系統(tǒng)彈性變形，工藝系統(tǒng)的振動，刀具耐用度，機床功率等。在綜合考慮上述因素的基礎(chǔ)上，使背吃刀量ap，進給量f ，切削速度v的積最大。一般應(yīng)先盡量取在的ap，其次盡量取大的進給量f，最后取合適的切削速度 2.8加工工序綜合卡 浙江紡織服裝學(xué)院 傳動齒輪加工工序卡片 產(chǎn)品類型 共1頁 零件名稱 傳動齒輪 第1頁 毛坯材料 20CrMnTi 毛坯種類 鍛件 每毛坯件數(shù) 1 工序號 工序名稱 工序內(nèi)容 車間 加工設(shè)備 夾具 備注 01 鍛 鍛造毛坯 鍛 02 熱處理 對毛坯進行正火處理 熱 03 車 粗車Φ35內(nèi)孔，留磨削余量0.4mm 機加 車床 三爪卡盤 04 車 粗車Φ92.55外圓，端面 機加 車床 專用夾具 05 車 半精車外圓。留磨削余量0.4mm 機加 車床 專用夾具 06 車 齒端的加工倒圓，倒尖，倒棱 機加 車床 專用夾具 07 滾齒 輪齒齒形的滾齒加工 機加 滾齒機 專用夾具 08 熱處理 齒面的滲碳淬火熱處理58~64HRC 熱 09 插鍵槽 插8*3.3的鍵槽。保證和A0.035的平行度與0.05的對稱度 機加 插床 三爪卡盤 10 倒角 倒角1×45°，1.5×45° 11 磨 外圓至尺寸 機加 磨床 心棒 12 磨 磨內(nèi)孔至Φ35mm公差要求0~+0.027 機加 磨床 三爪卡盤 13 磨 齒面磨削 機加 磨齒機 專用夾具 14 鉗 去毛刺 15 檢 檢驗 16 油封 清理，入庫 ` 編制（日期） 審核（日期） 會簽（日期） 標(biāo)記 文件號 簽字 日期 3、夾具設(shè)計 1.確定夾具的類型 由齒輪零件的加工比較簡單, 主要是齒輪端面的加工,故可采用心軸定位,再用螺栓緊固零件來定位夾緊工件.夾具的簡圖如下: 1.確定工件的定位方式及定位元件的結(jié)構(gòu) 工件的定位方式主要取決于工件的加工要求和定位基準(zhǔn)的形狀．尺寸。分析加工工序的技術(shù)條件和定位基準(zhǔn)選擇的合理性，遵循六點定位原則，按定位可靠．結(jié)構(gòu)簡單的原則，確定定位方式。常見的定位方式有平面定位．內(nèi)孔定位．外圓定位和組合定位（一孔一端面）定位等。在確定了工件的定位方式后，即可根據(jù)定位基面的形狀，選取相應(yīng)的定位元件及結(jié)構(gòu)。 2.工件的夾緊方式，計算夾緊力并設(shè)計夾緊裝置 夾緊機構(gòu)應(yīng)保證工件夾緊可靠．安全．不破壞工件的定位及夾壓表面的精度和粗糙度。在設(shè)計夾緊裝置時必須合理選擇夾緊力的方向和作用點，必要時還應(yīng)進行夾緊力的估算。 在確定夾緊力的大小時，為簡化計算，通常將夾具和工件看成一個剛性系統(tǒng)。根據(jù)工件所受切削力，夾緊力（大型工件還需考慮重力，慣性力等等）的作用情況，找出加工過程中對夾緊最不力的狀態(tài)，按靜力平衡原理計算出理論夾緊力，最后再乘以安全系數(shù)， 如下： 用拉桿壓板夾緊工件端面 2KM Fwk=（d+D）f Fwk為所需（實際）夾緊力； M為切削轉(zhuǎn)矩； K為安全系數(shù)(粗加工時取2.5～3，精加工時取1.5～2)； d為工件直徑； f為工件與支承面間的摩擦系數(shù)； 3.2夾緊力的計算 (1)夾緊力的計算 2KM 查表得Fwk=（d+D）f 取K=2，f=0.2 得Fwk=774.9(N) (2)選擇夾緊螺栓直徑 根據(jù)夾緊力方向和作用點的選擇原則，選用兩只M16×60內(nèi)角夾緊螺栓，加工時縱向進刀時，軸向力Fz與夾緊力Fwk方向相同，切削扭矩M切使得工件轉(zhuǎn)動，為防止工件發(fā)生轉(zhuǎn)動，夾具夾緊機構(gòu)應(yīng)有足夠的摩擦力矩與之平衡,選用M16的螺栓作為夾緊螺栓，安全可靠。 3.3夾具設(shè)計圖 設(shè)計小結(jié) 通過傳動齒輪畢業(yè)設(shè)計（包括工藝設(shè)計和夾具設(shè)計），它的全面性和系統(tǒng)性，使我對機械加工產(chǎn)生了更進一步的興趣，并讓我主觀上認識了加工過程中如何找定位，從而為傳動齒輪中重要尺寸（包括尺寸精度和位置精度）來滿足實際中與之相配套的用途。不僅是齒輪類的，還有其它類型，在設(shè)計時都為加工出更好使用的性能。在設(shè)計時通過查閱資料使得了解的知識面更廣、范圍更大，在查閱時了解有關(guān)機械方面的書，從而切實地感受到機械行業(yè)的范圍涉及面之廣，它能指導(dǎo)實際，理論知識有待實際的檢驗，理論知識與實際操作是相輔相成。 . 致 謝 在這本設(shè)計即將完成之際,我要感謝一直幫助我的楊偉超、繆飛軍兩位導(dǎo)師，他們在我的設(shè)計中給了我很多的指導(dǎo)，在一些對我來說比較艱深，比較把握不了方向的地方，給予了我指明，幫我除去了許多不必要的麻煩；在此我還要感謝我們公司技術(shù)課的同仁，是他們不遺余力的幫助，才使我有充足的時間來完成本設(shè)計。謝謝你們，因為有你們才有本設(shè)計的實現(xiàn)。 參考文獻 機械制造技術(shù)課程設(shè)計 /吳雄彪主編、—杭州：浙江大學(xué)出版社，2005、1 機械制造基礎(chǔ) /蘇建修主編、—北京：機械工業(yè)出版社，2001、5 工程材料 /許德珠主編、—2版、—北京：高等教育出版社，2001、6 機床夾具設(shè)計手冊 /東北重型機械學(xué)院，洛陽工學(xué)院，第一汽車制造廠職工大學(xué)編、—上海：上?？茖W(xué)技術(shù)出版社，1990 切削用量簡明手冊 /艾興，肖詩綱編、—北京：機械工業(yè)出版社，1993 機械制造工藝學(xué) /徐嘉元，曾家駒主編，機械制造工藝學(xué)、—北京：機械工業(yè)出版社，1998 機械設(shè)計手冊 /徐灝主編、—北京：機械工業(yè)出版社，1991 機械加工工藝手冊 /孟少農(nóng)主編、—北京：機械工業(yè)出版社，1991 機械工程手冊/張志仁主編、—北京：機械工業(yè)出版社，2000 機械制造技術(shù)課程設(shè)計/吳雄彪、—浙江大學(xué)出版社，2005 浙 江學(xué) 院 畢業(yè)設(shè)計----傳動齒輪工藝設(shè)計 設(shè)計人： 班 級： 學(xué) 號： 指導(dǎo)老師： 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