濾油器體工藝及鉆11斜出油孔夾具設計【含CAD圖紙、工藝工序卡、說明書】

資源目錄里展示的全都有，所見即所得。下載后全都有,請放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 XX大學課程設計濾油器體工藝夾具設計21目錄1 零件的分析及生產(chǎn)類型的確定21.1零件的作用21.2 零件的工藝分析31.3 零件的生產(chǎn)類型32 零件毛坯的設計32.1 選擇毛坯32.2 毛坯尺寸公差與機械加工余量的確定42.3 確定毛坯尺寸42.4 設計毛坯圖63 零件的加工工藝設計73.1 定位基準的選擇73.2 零件表面加工方法的選擇73.3 擬訂工藝路線73.4 工藝方案的比較與分析93.5 工序設計103.5.1 選擇加工設備與工藝裝備103.5.2 確定工序尺寸124 夾具的設計154.1 工件的定位154.2 夾緊裝置174.3 定位誤差分析174.4 對刀裝置184.5 設計小結18參考文獻201 零件的分析及生產(chǎn)類型的確定1.1零件的作用“CA6140車床濾油器體”如圖1所示。它位于車床主軸箱上圖1：CA6140車床濾油器體零件圖面，主要作用是給主軸箱內(nèi)供油及對油液起沖作用，零件的兩段有兩孔用于油液的進出，零件的中間有一個48h6的外圓柱面，用于與主軸箱以基軸制形式聯(lián)接。1.2 零件的工藝分析 “CA6140車床濾油器體”的各表面： （1）、零件的左端面（用于精基準加工其他表面）； （2）、螺栓孔3-9（用于聯(lián)接車床主軸箱，起固定作用）；（3）、中心孔38（用于過濾及緩沖油液）；（4）、進出油孔2-11（用于聯(lián)接進出油裝備，流通油液）；（5）、外圓柱面48h6（用于與車床主軸箱聯(lián)接）。 各表面的相互精度要求有：（1）、外圓柱面48h6為基軸制聯(lián)接，尺寸精度為IT6；（2）、其他表面無特殊精度要求，除保證其表面粗糙度外，尺寸精度為IT14。1.3 零件的生產(chǎn)類型依設計題目知：Qn=8000件/年；結合生產(chǎn)實際，備品率和廢品率可以取為=5%,=0.5%。由此可得，該零件的生產(chǎn)綱領 C6140車床濾油器體的質量為1.1kg,查表可知其屬輕型零件，生產(chǎn)類型為中大批量生產(chǎn)。2 零件毛坯的設計2.1 選擇毛坯根據(jù)生產(chǎn)綱領可知，CA6140濾油器體屬中大批量生產(chǎn)，零件形狀為非全圓柱體，可選零件材料為灰口鑄鐵，毛坯制造選用鑄造毛坯，這樣毛坯與成品相似，加工方便，省工省料。為了提高生產(chǎn)率，鑄造方法選用砂型鑄造，且為機器造型。2.2 毛坯尺寸公差與機械加工余量的確定 （1）、求最大輪廓尺寸 根據(jù)零件圖輪廓尺寸，可知，CA6140濾油器體最大輪廓尺寸為102mm。 （2）、選取公差等級CT 鑄造方法按機器造型、鑄件材料按灰鑄鐵，查表得，公差等級CT范圍是812級，取為10級。 （3）、求鑄件尺寸公差 根據(jù)加工面的基本尺寸和鑄件公差等級CT，查表得，公差帶相對與基本尺寸對稱分布。 （4）、求機械加工余量等級 鑄造方法按機器造型、鑄件材料按灰鑄鐵，查表得，機械加工余量等級范圍是EG級，取為F級。、（5）、求RAM（要求的機械加工余量） 對所有的加工表面取同一個數(shù)值，由最大輪廓尺寸102mm、機械加工余量等級為F級，得RAM數(shù)值為1.5mm。 2.3 確定毛坯尺寸上面查得的加工余量適用于機械加工表面粗糙度Ra1.6m。Ra1.6m的表面，余量要適當增大。分析零件，各加工表面均為Ra1.6m，因此這些表面的毛坯尺寸只需要將零件的尺寸加上余量值即可。零件上孔3-9和進出有口尺寸較小，鑄成實心；A面為單側加工，則 38盲孔屬內(nèi)腔加工，根據(jù)零件分析，38深度為65不變，徑向為 30盲孔屬內(nèi)腔，且此孔為非配合孔，用鉆床锪出即可，所以30盲孔鑄成實心。48h6外圓柱面的加工，即 如圖2所示，由于38孔的深度和三角形臺階肩寬度的余量和A面余量有關，A面增量a=105.3-102=3.3mm;所以38孔的毛坯深度=65+3.3=68.3mm;三角形臺階肩寬度=34+3.3=37.3mm。圖2：根據(jù)零件尺寸計算的毛坯尺寸綜合上述，確定毛坯加工部分的尺寸，見表1所示。表1：CA6140車床濾油器體毛坯（鑄件）尺寸。單位：mm。項目A面38徑向38軸向48徑向三角肩寬3-92-1130公差等級CT1010101010加工面基本尺寸10238654834鑄件尺寸公差3.62.6同A面2.8同A面機械加工余量等級FF同A面F同A面RMA1.51.5同A面1.5同A面毛坯基本尺寸105.333.768.352.437.30002.4 設計毛坯圖（1） 、確定圓角半徑 外圓角半徑 r=1mm; 內(nèi)圓角半徑 R=1mm; 以上所取圓角半徑數(shù)值能保證各表面的加工余量。（2）、確定拔模斜度 本鑄件最大尺寸為h=105.3mm，屬25500mm的鑄件，所以查表，取拔模斜度為1：20。（3） 、確定分型面由鑄件結構分析，選擇左端面作為砂模鑄造機器造型的分型面。 （4）、確定毛坯的熱處理方式 灰鑄鐵濾油器毛坯鑄造后應安排人工時效，溫度，進行消除殘余應力，從而改善加工性。綜合上述，所設計的毛坯圖如圖3所示。圖3： CA6140車床濾油器體毛坯3 零件的加工工藝設計3.1 定位基準的選擇本零件是不規(guī)則多孔零件體，其左端面是設計基準（亦是裝配基準和測量基準），為了避免由于基準不重合而產(chǎn)生的誤差，應選左端面為定位基準，即遵循“基準重合”的原則。 在加工中還可遵循“互為基準”的原則，重復加工38孔和48外圓柱面，使其中心線重合度較高，減小圓跳動度。3.2 零件表面加工方法的選擇本零件有平面、內(nèi)孔、外圓柱面、螺紋等加工，材料為HT15-33。以公差等級和表面粗糙度要求，參考本指南有關資料，其加工方法選擇如下：（1） 、左端面 左端面為精基準加工其它表面，表面粗糙度為Ra1.6，加工方法為先車后磨。 （2）、38mm內(nèi)孔 未注公差尺寸，根據(jù)GB1800-79規(guī)定其公差等級按IT14，表面粗糙度為Ra6.3，由于毛坯上已經(jīng)鑄造出孔留出加工余量，所以選擇先擴孔再鉸孔即可達到要求。 （3）、3-9孔 未注公差尺寸，根據(jù)GB1800-79規(guī)定其公差等級按IT14，表面粗糙度為Ra6.3，只需用9mm麻花鉆直接在其位置鉆孔即可。 （4）、48外圓柱面 公差等級為IT6，表面粗糙度為Ra1.6，加工方法可采用粗車精車精磨。 （5）、進出油孔（2-11） 這兩個孔用于聯(lián)接油管，用于輸通油液，密封性需要比較高。連結部分采用螺紋聯(lián)接。由于毛坯中這兩孔是鑄為實心，所以此部分的加工為先锪沉頭孔，然后在其位置鉆11孔，接著擴孔到16深18mm，最后攻絲M18。3.3 擬訂工藝路線 制定工藝路線的出發(fā)點，應當是使其零件的幾何形狀，尺寸精度及位置精度等技術要求能得到合理的保證，在生產(chǎn)綱領中已經(jīng)確定為成批生產(chǎn)的條件下，可以考慮采用通用機床配以專用夾具，除此之外，應當考慮經(jīng)濟效果，以便使生產(chǎn)成本盡量下降。 濾油器體加工包括各圓柱面、孔及端面的加工。按照先加工基準面及先粗后精的原則，該零件加工可按下述工藝路線進行。（1） 、工藝路線方案一 工序 機器砂型鑄造毛坯； 工序 檢驗，清砂； 工序 熱處理； 工序 磨左端面； 工序 鉆38內(nèi)孔及3-9孔； 工序 粗車外圓48； 工序 加工出油口； 工序 加工進油口； 工序 磨48外圓柱面； 工序 檢驗。（2）、工藝路線方案二工序 機器砂型鑄造毛坯； 工序 檢驗，清砂； 工序 熱處理； 工序 磨左端面； 工序 車48外圓柱面； 工序 磨48外圓柱面； 工序 鉆3-9孔； 工序 銑30內(nèi)孔面； 工序 擴、鉸38孔； 工序 38孔的倒角； 工序 锪平26； 工序 加工進油口； 工序 加工出油口； 工序 檢驗。 （3）工藝路線方案三工序 機器砂型鑄造毛坯； 工序 檢驗毛坯、清砂； 工序 熱處理； 工序 粗車左端面； 工序 先锪30內(nèi)孔面，再擴38內(nèi)孔； 工序 粗車48外圓，切退刀槽； 工序 鉸38內(nèi)孔，內(nèi)孔倒角； 工序 精車48外圓； 工序 鉆3-9通孔； 工序 磨左端面； 工序 加工出油孔； 工序 加工進油孔； 工序 去毛刺； 工序 精磨48外圓； 工序 檢驗。3.4 工藝方案的比較與分析 上述三個工藝方案的特點在于：方案一，將工序集中，但需要專用機床才能達到加工要求。方案二，工序分散，但工藝按排的加工達不到精度要求。方案三，也是分散工序，工藝安排合理，能滿足用通用機床加工并達到精度的要求，除了選擇萬能性通用機床加工外，還要設計一些專用夾具，提高加工要求和質量。因此選用第三個方案較合理，具體如下：工序 機器造型砂模鑄造毛坯； 工序 清砂，檢驗毛坯各尺寸，不得有砂眼缺陷； 工序 熱處理，人工時效溫度，消除殘余應力； 工序 夾外圓，粗車左端面； 工序 先锪30內(nèi)孔面，再擴38內(nèi)孔； 工序 粗車48外圓，切退刀槽； 工序 鉸38內(nèi)孔，內(nèi)孔倒角； 工序 精車48外圓； 工序 鉆3-9通孔； 工序 磨左端面； 工序 加工出油孔（锪26平面，鉆11孔，擴16孔深18mm,攻絲M181.5深14mm）； 工序 加工進油孔（锪26平面，鉆11孔，擴16孔深18mm,攻絲M181.5深12mm）； 工序 用鉗工去毛刺； 工序 精磨48外圓； 工序 檢驗，是否達到要求的精度和粗糙度。3.5 工序設計3.5.1 選擇加工設備與工藝裝備 （1）、選擇機床 根據(jù)不同的工序選擇機床。 工序、 屬于備坯階段，不屬于切削加工，由專門的 車間負責。 工序 粗車左端面，選用臥式車床就能滿足要求。本零件外廓尺寸不大，精度要求不是很高，根據(jù)工廠內(nèi)的現(xiàn)有設備一般最常用CA6140型臥式車床加工。 工序 先锪30內(nèi)孔面，再擴38內(nèi)孔，精度要求不高，選用立式鉆床進行加工。 工序 粗車48外圓柱面，在切退刀槽。加工中需要換刀具，刀架最好是換位的，所以選擇CA6140最經(jīng)濟。 工序 鉸38內(nèi)孔，內(nèi)孔倒角，選鉆床進行加工。 工序 精車48外圓，選用CA6140進行加工。 工序 鉆3-9通孔，由于是通孔，刀具進給深度可以不嚴格控制，在其位置鉆通孔即可。選用立式加工中心，只要在專用夾具上設計好對刀塊，然后執(zhí)行程序就可以加工出3個9通孔了。 工序x 磨左端面，表面粗糙度Ra1.6mm，沒有要求與中心線的垂直度，所以選用普通的端面磨床即可加工達到要求。 工序、加工進（出）油口孔，選用搖臂鉆床進行加工。 工序 精磨48外圓，公差等級為IT6，表面粗糙度為Ra1.6mm，精度要求比較高，所以這里選用外圓磨床進行加工。（2）、選擇夾具 工序 選用三爪自定心卡盤。工序、 都選用專用夾具。（3）選擇刀具 根據(jù)不同的工序選擇刀具。在車床上加工的工序 一般選用硬質合金車刀。切刀槽宜選用高速鋼。為提高生產(chǎn)率及經(jīng)濟性，應選用可轉位車刀架。在鉆床上機加工的工序 麻花鉆，锪頭鉆，倒孔鉆等。（4） 選擇量具 加工部分的量具工序 卡尺； 工序 卡尺； 工序 卡尺； 工序 卡尺；工序 卡尺； 工序 卡尺； 工序 卡尺； ； 工序 卡尺，螺紋塞規(guī)； 工序 卡尺，螺紋塞規(guī)； 工序 千分尺。 檢驗部分的量具 檢查內(nèi)容 測量工具砂眼、清砂是否干凈、裂縫 觀察表面及磁力探傷；軸向尺寸 卡尺；外圓徑向尺寸 外徑千分尺；角度 角度卡尺；孔內(nèi)徑 內(nèi)徑千分尺； 檢查粗糙度 專用儀器； 檢查配合表面精度 專用儀器。綜合上述，CA6140車床濾油器體的工藝過程如表2所示。表2：CA6140車床濾油器體的工藝過程及設備工序號工序內(nèi)容機床備注00機器砂型鑄造毛坯附表205檢驗毛坯、清砂附表310熱處理附表420粗車左端面CA6140附表530先锪30內(nèi)孔面，再擴38內(nèi)孔立式鉆床附表640粗車48外圓，車退刀槽CA6140附表750鉸38內(nèi)孔，倒角搖臂鉆床附表860精車48外圓CA1640附表970鉆3-9通孔立式加工中心附表1080磨左端面端面磨床附表1190加工出油孔搖臂鉆床附表12100加工進油孔搖臂鉆床附表13110去毛刺附表14120精磨48外圓外圓磨床附表15130檢驗附表16 3.5.2 確定工序尺寸 根據(jù)各原始資料及制定的零件加工工藝路線，采用計算與查表相結合的方法確定各工序加工余量，中間工序公差按經(jīng)濟精度選定，上下偏差按入體原則標注，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： “CA6140車床濾油器體”零件材料為灰口鑄鐵，硬度HB207233，毛坯的重量約為1.1千克。生產(chǎn)類型為中批或大批，采用機械砂型鑄造毛坯。（1）、確定圓柱面的工序尺寸38，30，11，3-9由于無特殊加工精度等，在滿足其表面粗糙度的條件下直接鉆孔或擴孔來完成。外圓柱面46h6和46： 46粗糙度為Ra6.3,精度等級為IT14，考慮其長度為14mm,因為砂型鑄造等級為IT14左右，能滿足加工需求，因此直接鑄造成型。 48h6所需精度要求高，需進行粗車精車精磨。精車此時直徑余量為0.9mm，精磨此時直徑余量為0.1mm， 剩下的粗車時直徑余量為52.4-48-0.9=3.4mm，能滿足加工要求，考慮48h6長度為40，精度等級為14，由鑄造直接完成。48h6的外圓柱面的加工余量表如下。表3:48h6外圓柱面的加工工序尺寸（單位：mm）工序名稱工序雙邊余量工序公差工序基本尺寸工序尺寸及公差精磨0.1IT6(0-0.016)48精車0.9IT7(0-0.025)48.1粗車3.4IT11(0-0.160)49毛坯F(0-3.6)52.4(2)確定軸向工序尺寸 綜合上述，加工的工序尺寸如表4所示。表4：各加工工序的工序尺寸工序工位工位尺寸I鑄造毛坯機器砂型鑄造毛坯參見毛坯圖所示II備坯檢驗毛坯、清砂熱處理人工時效車左端面粗車左端面保長102.5mm锪30內(nèi)孔，擴38內(nèi)孔先锪30內(nèi)孔平面锪30內(nèi)孔保長70.5mm再擴38內(nèi)孔擴內(nèi)孔至37，保長65.5mm車48外圓和退刀槽粗車48外圓車外圓至49車退刀槽3x1.5mm鉸38內(nèi)孔和倒角鉸38內(nèi)孔鉸內(nèi)孔至38，保長65.5mm內(nèi)孔倒角1C精車48外圓精車48外圓精車外圓至48.1鉆3-9通孔鉆3-9通孔鉆3-9通孔磨左端面磨左端面保長102mm加工出油口空口锪平锪平26鉆孔鉆11孔深32mm擴孔擴孔至16孔，深18mm攻絲攻絲M18x1.5，深14加工進油口孔口锪平锪平26鉆孔鉆11孔通至下一平面擴孔擴孔至16孔 8mm攻絲攻絲M18x1.5，深12XIII去毛刺去毛刺X精磨48外圓精磨48外圓精磨外圓至48X檢驗檢驗4 夾具的設計本夾具是工序XII用11麻花鉆直接鉆斜孔的專用夾具，在Z39搖臂鉆床是加工出11斜孔（通孔）。所設計的夾具裝配圖如附圖3所示，夾具體零件圖如附圖4所示。有關說明如下：4.1 工件的定位（1）、定位方案 所加工的11孔是一斜孔，本工序加工要求保證的位置精度主要是11中心線與26沉頭孔平面的交點F到A面的尺寸94IT14；即。根據(jù)基準重合原則，應選端面A為主要定位基準。該工序鉆斜孔時受力不大，但受力不平衡，所以要實現(xiàn)完全定位，在此可以用“一面兩孔”定位原則，已知端面A為主要定位基準，然圖6：工件的定位與夾緊后可以再選38孔和下面的9孔作為定位孔。端面A可以限制3個自由度，38孔與定位銷配合可以限制2個自由度，9孔與定位銷配合可限制1個自由度，即實現(xiàn)完全定位。（2） 計算定位銷尺寸 （a） (b)圖7：定位銷 38孔選用圓柱銷定位，9孔選用菱形銷定位。下面只計算定位銷的配合尺寸，其余尺寸按標準查表。、確定兩銷中心距 按IT14查表，得9孔的公差為0.36mm，38孔的公差為0.62mm。取孔的中心偏差等于孔的公差的1/5，則9孔的中心偏差為，38孔的中心偏差為。那么兩孔中心距偏差為。兩孔偏差較大，那么兩銷中心距的偏差為。、確定圓柱定位銷直徑尺寸 取 、確定菱形銷寬度b和B值 如圖7（b）所示的菱形銷。查表可得 b=4mm ， B=7mm計算菱形銷的直徑尺寸 取偏差為g6，查表得mm上述式中的：削邊銷與孔配合的最小間隙，單位為mm;b削邊銷的寬度，單位為;分別為工件上兩孔中心距公差和夾具上兩銷中心距公差，單位為mm；工件上削邊銷定位孔直徑，單位為mm；削邊銷直徑尺寸，單位為mm。4.2 夾緊裝置入圖8所示，這是專門設計的夾緊元件。參見夾具裝配圖，把夾緊元件套入零件的外圓柱面，然后順時針旋轉將套筒的兩翼卡入夾具體的契形口中，契形口是按一定角度設計的，可讓兩翼受力時產(chǎn)生摩擦力而自鎖，達到夾緊作用。圖8：契形雙翼套筒4.3 定位誤差分析如圖6所示。所加工的是11斜孔，本工序加工要求保證的位置精度主要是11孔中心線與沉頭孔平面的交點到左端面A的尺寸。由于左端面A是主要定位基準，那么定位基準與工序基準重合；由于定位副制造誤差引起的定位誤差是受兩定位銷影響的；在左端面A平面方向移動的定位誤差由38圓柱銷確定，由于圓柱銷與38孔為過盈配合，所以。繞端面A的垂直線旋轉方向的定位誤差由9菱形銷確定，9孔的尺寸為，菱形銷為mm，那么。由上述得，由三角函數(shù)關系換算成尺寸方向得。那么，該方案的定位誤差小于該工序尺寸制造公差087mm的1/3 （），該定位方案可行。4.4 對刀裝置如裝配圖所示，由專用夾具的定位與夾緊，11變?yōu)樨Q直孔，只要讓鉆頭豎直進給加工就可以了。由于零件是斜著安裝到夾具上的，為了方便裝卸工件，把鉆模板設計成鉸鏈翻開式的。已知生產(chǎn)類型為中大批量生產(chǎn)，而且鉆好11孔后好要進行擴孔與攻絲，就需要更換刀具，所以選用快換鉆套比較合適。2、 夾具體 工件的定位是由夾具體的斜面和兩個可換定位銷連接起來的。然后通過雙翼套筒卡在夾具體的契形口里實現(xiàn)夾緊。通過鉆模板連接套筒和夾具體，實現(xiàn)對刀。這樣該夾具便有機連接起來，實現(xiàn)了定位、夾緊、對刀等功能。3、 結構特點 該夾具結構簡單，操作方便。 夾緊是通過雙翼套筒的契形自鎖實現(xiàn)的，而鉆模板用鉸鏈翻開式，解決了空間問題，使工件裝卸變得方便。 本夾具更換鉆套后，又可以進行16擴孔和M18X1.5mm攻絲的工位。4.5 設計小結這是我大學期間，在學完全部課程和生產(chǎn)實習之后的又一次實踐性學習。經(jīng)過了3個星期努力，我終于完成了任務。在這次畢業(yè)設計中，使我懂得了運用學過的知識，把理論融入到實踐中。以前在課本中不明白的很多問題，也得到了徹底的解決。這次畢業(yè)設計主要是根據(jù)要求設計工藝和某道工序的專用夾具。設計工藝與設計夾具，我都是第一次實踐，經(jīng)過老師的指導，我都順利完成了。在畢業(yè)設計中，我懂得了解決問題并不是只有一種方法的道理，要實事求是，要以花錢少辦好事為原則，這樣才能成為一名優(yōu)秀的設計人員。在此，感謝學校和各位老師給了我這個難得的學習機會，我將會更努力地完成我的學業(yè)，為校爭光！參考文獻機械制造技術基礎畢業(yè)設計指南主編：宗凱 化學工業(yè)出版社機械制造工藝學主編：陳明 機械工業(yè)出版社機械制造技術基礎畢業(yè)設計指導教程主編：鄒青 機械工業(yè)出版社 機械設計畢業(yè)設計手冊主編：吳宗澤、羅圣國 高等教育出版社Robotics and Computer-Integrated Manufacturing 21 (2005) 368378Locating completeness evaluation and revision in fixture planH. Song?, Y. RongCAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USAReceived 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004AbstractGeometry constraint is one of the most important considerations in fixture design. Analytical formulation of deterministiclocation has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process ofactual fixture design practice has not been thoroughly studied. In this paper, a methodology to characterize fixturing systemsgeometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognizedwith given locating scheme. All un-constrained motions of a workpiece in an under-constraint status can be automatically identified.This assists the designer to improve deficit locating scheme and provides guidelines for revision to eventually achieve deterministiclocating.r 2005 Elsevier Ltd. All rights reserved.Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained1. IntroductionA fixture is a mechanism used in manufacturing operations to hold a workpiece firmly in position. Being a crucialstep in process planning for machining parts, fixture design needs to ensure the positional accuracy and dimensionalaccuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a locationscheme. V-block and pin-hole locating principles are also commonly used.A location scheme for a machining fixture must satisfy a number of requirements. The most basic requirement is thatit must provide deterministic location for the workpiece 1. This notion states that a locator scheme producesdeterministic location when the workpiece cannot move without losing contact with at least one locator. This has beenone of the most fundamental guidelines for fixture design and studied by many researchers. Concerning geometryconstraint status, a workpiece under any locating scheme falls into one of the following three categories:1. Well-constrained (deterministic): The workpiece is mated at a unique position when six locators are made to contactthe workpiece surface.2. Under-constrained: The six degrees of freedom of workpiece are not fully constrained.3. Over-constrained: The six degrees of freedom of workpiece are constrained by more than six locators.In 1985, Asada and By 1 proposed full rank Jacobian matrix of constraint equations as a criterion and formed thebasis of analytical investigations for deterministic locating that followed. Chou et al. 2 formulated the deterministiclocating problem using screw theory in 1989. It is concluded that the locating wrenches matrix needs to be full rank toachieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. 3 consideredARTICLE IN PRESS front matter r 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.rcim.2004.11.012?Corresponding author. Tel.: +15088316092; fax: +15088316412.E-mail address: hsongwpi.edu (H. Song).locatorworkpiece contact area effects instead of applying point contact. They introduced a contact matrix andpointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson 4suggested that a linear approximation may not be sufficient for some applications such as non-prismatic surfaces ornon-small relative errors. He proposed a second-order Taylor expansion which also takes locator error interaction intoaccount. Marin and Ferreira 5 applied Chous formulation on 3-2-1 location and formulated several easy-to-followplanning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyzenon-deterministic location.In the Asada and Bys formulation, they assumed frictionless and point contact between fixturing elements andworkpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surfacefunction is gi(as shown in Fig. 1).The surface function is defined as giq? 0: To be deterministic, there should be a unique solution for the followingequation set for all locators.giq 0;i 1;2;.;n,(1)where n is the number of locators and q x0;y0;z0;y0;f0;c0? represents the position and orientation of theworkpiece.Only considering the vicinity of desired location q?; where q q? Dq; Asada and By showed thatgiq giq? hiDq,(2)where hiis the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locatingrequirement can be satisfied if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solutionq q?:rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0:qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0:qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc026666666664377777777758:9=; 6.(3)Upon given a 3-2-1 locating scheme, the rank of a Jacobian matrix for constraint equations tells the constraint statusas shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one freemotion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme hasmore than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that itcan be removed without affecting the geometry constrain status of the workpiece.For locating a model other than 3-2-1, datum frame can be established to extract equivalent locating points. Hu 6has developed a systematic approach for this purpose. Hence, this criterion can be applied to all locating schemes.ARTICLE IN PRESSX Y Z O X Y Z O (x0,y0,z0) gi UCS WCS Workpiece Fig. 1. Fixturing system model.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378369Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module intheir automated computer-aided fixture design verification system. Their CAFDV system can calculate the Jacobianmatrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivityto locating error.Xiong et al. 8 presented an approach to check the rank of locating matrix WL(see Appendix). They also intro-duced left/right generalized inverse of the locating matrix to analyze the geometric errors of workpiece. It hasbeen shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators arerelated as follows:Well-constrained :DX WLDr,(4)Over-constrained :DX WTLWL?1WTLDr,(5)Under-constrained :DX WTLWLWTL?1Dr I6?6? WTLWLWTL?1WLl,(6)where l is an arbitrary vector.They further introduced several indexes derived from those matrixes to evaluate locator configurations, followed byoptimization through constrained nonlinear programming. Their analytical study, however, does not concern therevision of non-deterministic locating. Currently, there is no systematic study on how to deal with a fixture design thatfailed to provide deterministic location.2. Locating completeness evaluationIf deterministic location is not achieved by designed fixturing system, it is as important for designers to knowwhat the constraint status is and how to improve the design. If the fixturing system is over-constrained, informa-tion about the unnecessary locators is desired. While under-constrained occurs, the knowledge about all the un-constrained motions of a workpiece may guide designers to select additional locators and/or revise the locatingscheme more efficiently. A general strategy to characterize geometry constraint status of a locating scheme is describedin Fig. 2.In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status (see Appendixfor derivation of locating matrix). The deterministic locating requires six locators that provide full rank locatingmatrix WL:As shown in Fig. 3, for given locator number n; locating normal vector ai;bi;ci? and locating position xi;yi;zi? foreach locator, i 1;2;.;n; the n ? 6 locating matrix can be determined as follows:WLa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775.(7)When rankWL 6 and n 6; the workpiece is well-constrained.When rankWL 6 and n46; the workpiece is over-constrained. This means there are n ? 6 unnecessary locatorsin the locating scheme. The workpiece will be well-constrained without the presence of those n ? 6 locators. Themathematical representation for this status is that there are n ? 6 row vectors in locating matrix that can be expressedas linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist oneARTICLE IN PRESSTable 1RankNumber of locatorsStatuso 6Under-constrained 6 6Well-constrained 646Over-constrainedH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370locating scheme that provides deterministic location. The developed algorithm uses the following approach todetermine the unnecessary locators:1. Find all the combination of n ? 6 locators.2. For each combination, remove that n ? 6 locators from locating scheme.3. Recalculate the rank of locating matrix for the left six locators.4. If the rank remains unchanged, the removed n ? 6 locators are responsible for over-constrained status.This method may yield multi-solutions and require designer to determine which set of unnecessary locators shouldbe removed for the best locating performance.When rankWLo6; the workpiece is under-constrained.3. Algorithm development and implementationThe algorithm to be developed here will dedicate to provide information on un-constrained motions of theworkpiece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ARTICLE IN PRESSFig. 2. Geometry constraint status characterization.X Z Y (a1,b1,c1) 2,b2,c2) (x1,y1,z1) (x2,y2,z2) (ai,bi,ci) (xi,yi,zi) (aFig. 3. A simplified locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378371orientation errors and locator errors can be expressed as follows:DX DxDyDzaxayaz2666666666437777777775w11:w1i:w1nw21:w2i:w2nw31:w3i:w3nw41:w4i:w4nw51:w5i:w5nw61:w6i:w6n2666666666437777777775?Dr1:Dri:Drn2666666437777775,(8)where Dx;Dy;Dz;ax;ay;azare displacement along x, y, z axis and rotation about x, y, z axis, respectively. Driisgeometric error of the ith locator. wijis defined by right generalized inverse of the locating matrix Wr WTLWLWTL?15.To identify all the un-constrained motions of the workpiece, V dxi;dyi;dzi;daxi;dayi;dazi? is introduced such thatV DX 0.(9)Since rankDXo6; there must exist non-zero V that satisfies Eq. (9). Each non-zero solution of V represents an un-constrained motion. Each term of V represents a component of that motion. For example, 0;0;0;3;0;0? says that therotation about x-axis is not constrained. 0;1;1;0;0;0? means that the workpiece can move along the direction given byvector 0;1;1?: There could be infinite solutions. The solution space, however, can be constructed by 6 ? rankWLbasic solutions. Following analysis is dedicated to find out the basic solutions.From Eqs. (8) and (9)VX dxDx dyDy dzDz daxDax dayDay dazDaz dxXni1w1iDri dyXni1w2iDri dzXni1w3iDri daxXni1w4iDri dayXni1w5iDri dazXni1w6iDriXni1Vw1i;w2i;w3i;w4i;w5i;w6i?TDri 0.10Eq. (10) holds for 8Driif and only if Eq. (11) is true for 8i1pipn:Vw1i;w2i;w3i;w4i;w5i;w6i?T 0.(11)Eq. (11) illustrates the dependency relationships among row vectors of Wr: In special cases, say, all w1jequal to zero,V has an obvious solution 1, 0, 0, 0, 0, 0, indicating displacement along the x-axis is not constrained. This is easy tounderstand because Dx 0 in this case, implying that the corresponding position error of the workpiece is notdependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combinedmotion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. Forinstance, 9w1ja0;w2ja0; w1j ?w2jfor 8j: In this scenario, the workpiece cannot move along x- or y-axis. However, itcan move along the diagonal line between x- and y-axis defined by vector 1, 1, 0.To find solutions for general cases, the following strategy was developed:1. Eliminate dependent row(s) from locating matrix. Let r rank WL; n number of locator. If ron; create a vectorin n ? r dimension space U u1:uj:un?rhi1pjpn ? r; 1pujpn: Select ujin the way that rankWL r still holds after setting all the terms of all the ujth row(s) equal to zero. Set r ? 6 modified locating matrixWLMa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775r?6,where i 1;2;:;niauj:ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 3683783722. Compute the 6 ? n right generalized inverse of the modified locating matrixWr WTLMWLMWTLM?1w11:w1i:w1rw21:w2i:w2rw31:w3i:w3rw41:w4i:w4rw51:w5i:w5rw61:w6i:w6r26666666664377777777756?r3. Trim Wrdown to a r ? rfull rank matrix Wrm: r rankWLo6: Construct a 6 ? r dimension vector Q q1:qj:q6?rhi1pjp6 ? r; 1pqjpn: Select qjin the way that rankWr r still holds after setting all theterms of all the qjth row(s) equal to zero. Set r ? r modified inverse matrixWrmw11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r26666664377777756?6,where l 1;2;:;6 laqj:4. Normalize the free motion space. Suppose V V1;V2;V3;V4;V5;V6? is one of the basic solutions of Eq. (10) withall six terms undetermined. Select a term qkfrom vector Q1pkp6 ? r: SetVqk ?1;Vqj 0 j 1;2;:;6 ? r;jak;(5. Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found asfollows.v1:vs:v62666666437777775wqk1:wqki:wqkr2666666437777775?w11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r2666666437777775?1,where s 1;2;:;6saqj;saqk;l 1;2;:;6 laqj:6. Repeat step 4 (select another term from Q) and step 5 until all 6 ? r basic solutions have been determined.Based on this algorithm, a C+ program was developed to identify the under-constrained status and un-constrained motions.Example 1. In a surface grinding operation, a workpiece is located on a fixture system as shown in Fig. 4. The normalvector and position of each locator are as follows:L1:0, 0, 10, 1, 3, 00,L2:0, 0, 10, 3, 3, 00,L3:0, 0, 10, 2, 1, 00,L4:0, 1, 00, 3, 0, 20,L5:0, 1, 00, 1, 0, 20.Consequently, the locating matrix is determined.WL0013?100013?300011?20010?203010?2012666666437777775.ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378373This locating system provides under-constrained positioning since rankWL 5o6: The program then calculatesthe right generalized inverse of the locating matrix.Wr000000:50:5?1?0:51:50:75?1:251:5000:250:25?0:5000:5?0:50000000:5?0:526666666643777777775.The first row is recognized as a dependent row because removal of this row does not affect rank of the matrix. Theother five rows are independent rows. A linear combination of the independent rows is found according therequirement in step 5 of the procedure for under-constrained status. The solution for this special case is obvious that allthe coefficients are zero. Hence, the un-constrained motion of workpiece can be determined as V ?1; 0; 0; 0; 0; 0?:This indicates that the workpiece can move along x direction. Based on this result, an additional locator should beemployed to constraint displacement of workpiece along x-axis.Example 2. Fig. 5 shows a knuckle with 3-2-1 locating system. The normal vector and position of each locator in thisinitial design are as follows:L1:0, 1, 00, 896, ?877, ?5150,L2:0, 1, 00, 1060, ?875, ?3780,L3:0, 1, 00, 1010, ?959, ?6120,L4:0.9955, ?0.0349, 0.0880, 977, ?902, ?6240,L5:0.9955, ?0.0349, 0.0880, 977, ?866, ?6240,L6:0.088, 0.017, ?0.9960, 1034, ?864, ?3590.The locating matrix of this configuration isWL010515:000:8960010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:9960866:6257998:24660:093626666666643777777775,rankWL 5o6 reveals that the workpiece is under-constrained. It is found that one of the first five rows can beremoved without varying the rank of locating matrix. Suppose the first row, i.e., locator L1is removed from WL; theARTICLE IN PRESSXZYL3L4L5L2L1Fig. 4. Under-constrained locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374modified locating matrix turns intoWLM010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:996866:6257998:24660:09362666666437777775.The right generalized inverse of the modified locating matrix isWr1:8768?1:8607?20:666521:37160:49953:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775.The program checked the dependent row and found every row is dependent on other five rows. Without losinggenerality, the first row is regarded as dependent row. The 5 ? 5 modified inverse matrix isWrm3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:028402666666437777775.The undetermined solution is V ?1; v2; v3; v4; v5; v6?:To calculate the five undetermined terms of V according to step 5,1:8768?1:8607?20:666521:37160:499526666666643777777775T?3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775?1 0; ?1:713; ?0:0432; ?0:0706; 0:04?.Substituting this result into the undetermined solution yields V ?1;0; ?1:713; ?0:0432; ?0:0706; 0:04?ARTICLE IN PRESSFig. 5. Knuckle 610 (modified from real design).H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378375This vector represents a free motion defined by the combination of a displacement along ?1, 0, ?1.713 directioncombined and a rotation about ?0.0432, ?0.0706, 0.04. To revise this locating configuration, another locator shouldbe added to constrain this free motion of the workpiece, assuming locator L1was removed in step 1. The program canalso calculate the free motions of the workpiece if a locator other than L1was removed in step 1. This provides morerevision options for designer.4. SummaryDeterministic location is an important requirement for fixture locating scheme design. Analytical criterion fordeterministic status has been well established. To further study non-deterministic status, an algorithm for checking thegeometry constraint status has been developed. This algorithm can identify an under-constrained status and indicatethe un-constrained motions of workpiece. It can also recognize an over-constrained status and unnecessary locators.The output information can assist designer to analyze and improve an existing locating scheme.Appendix. Locating matrixConsider a general workpiece as shown in Fig. 6. Choose reference frame fWg fixed to the workpiece. Let fGg andfLig be the global frame and the ith locator frame fixed relative to it. We haveFiXw;Hw;rwi fiXli;Hli;rli,(12)where Xw2 3?1and Hw2 3?1(Xli2 3?1and Hli2 3?1) are the position and orientation of the workpiece(the ith locator) in the global frame fGg; rwi2 3?1(rli2 3?1) is the position of the ith contact point between theworkpiece and the ith locator in the workpiece frame fWg (the ith locator frame fLig).Assume that DXw2 3?1(DHw2 3?1) and Drwi2 3?1are the deviations of the position Xw2 3?1(orientationHw2 3?1) of the workpiece and the position of the ith contact point rwi2 3?1; respectively. Then we have the actualcontact on the wor