軸套鉆孔夾具設(shè)計(jì)【含CAD圖紙、SW三維】

資源目錄里展示的全都有，所見即所得。下載后全都有,請(qǐng)放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 夾具定位規(guī)劃中完整性評(píng)估和修訂 CAM實(shí)驗(yàn)室，機(jī)械工程學(xué)系，伍斯特理工學(xué)院研究院，100路，伍斯特，碩士01609，美國 2004年9月14日收稿;2004年11月9日修訂;2004年11月10日發(fā)表 摘 要 幾何約束是夾具設(shè)計(jì)中最重要的考慮因素之一。確定位置的解析擬訂已發(fā)達(dá)。然而，如何分析和修改在實(shí)際夾具設(shè)計(jì)實(shí)踐過程中的一個(gè)非確定性的定位計(jì)劃尚未深入研究。在本文中，提出了一種方法來描述在限制約束下的重點(diǎn)夾具系統(tǒng)的幾何約束狀態(tài)。一種限制約束下狀態(tài)，如果它存在，可以識(shí)別給定定位計(jì)劃?？梢宰詣?dòng)識(shí)別工件的所有限制約束下約束狀態(tài)的提案。這有助于改善逆差定位計(jì)劃，并為修訂提供指引，以最終實(shí)現(xiàn)確定性的定位。 關(guān)鍵詞：夾具設(shè)計(jì);幾何約束;確定性定位;限制約束;過約束 1.介紹 夾具是用于制造工業(yè)進(jìn)行工件牢固定位的一種機(jī)制。在零件加工過程中規(guī)劃一個(gè)關(guān)鍵的第一步，夾具設(shè)計(jì)需要，以確保定位精度和三維工件的精度。 3-2-1原則，在一般情況下，是最廣泛使用的指導(dǎo)原則發(fā)展的位置計(jì)劃。 V型塊和銷孔定位原則也常用。 一個(gè)加工夾具定位方案必須滿足一些要求。最基本的要求是，必須提供工件確定的位置。這種觀點(diǎn)指出，定位計(jì)劃生產(chǎn)的確定位置，工件不能移動(dòng)，而至少有一個(gè)定位不會(huì)失去聯(lián)系。這一直是夾具設(shè)計(jì)的最根本的準(zhǔn)則之一，許多研究人員關(guān)于幾何約束狀態(tài)的研究表明，工件在任何定位計(jì)劃分為以下三個(gè)類別： 1、良好的約束（確定性）：工件在一個(gè)獨(dú)特的位置進(jìn)行配合，工件表面與6個(gè)定位器取得聯(lián)系。 2、限制約束：不完全約束工件的自由度。 3、過約束：工件自由度超過6定位的制約。 在1985年,淺田[1]提出了滿秩為準(zhǔn)則雅可比矩陣的約束方程,基于分析形成了調(diào)研后,確定定位。周等[2]在1989年制定了在確定性定位問題上使用螺旋理論。結(jié)果表明,定位矩陣的定位需要壓力滿秩達(dá)到確定的位置。該方法的確定通過無數(shù)的研究。王等[3]考慮定位工件的接觸的影響，而采用點(diǎn)接觸面積。他們介紹了接觸矩陣，并指出，兩個(gè)接觸的機(jī)構(gòu)不應(yīng)該有平等的，但在接觸點(diǎn)曲率相反。卡爾森[4]認(rèn)為，可能沒有足夠的應(yīng)用，如一些不是非棱柱的表面或相對(duì)誤差近似的非小線性。他提出一個(gè)二階泰勒展開，其中也考慮到定位誤差相互作用。馬林和費(fèi)雷拉[5]應(yīng)用周對(duì)3-2-1的位置擬訂，制定若干按照規(guī)則的規(guī)劃。盡管眾多的位置上的確定分析研究很少注意非確定性分析的位置。 在淺田的擬定方案中,他們假設(shè)工件夾具元件和點(diǎn)之間的聯(lián)絡(luò)無阻力。理想的位置q*,而應(yīng)放置工件表面和分片，可微函數(shù)是gi(見圖1)。 表面函數(shù)定義為:gi(q*)=0是確定的,應(yīng)該有一個(gè)獨(dú)一無二的解決方案為下列所有定位方程組。 gi(q)=0,i=1,2,...,n (1) 其中n是定位器的位置與方向,代表了工件的定位和方向。 只有考慮到目標(biāo)位置q*附近在處: 淺田表明 (2) hi是幾何函數(shù)的雅可比矩陣，矩陣式所示（3）。確定定位 如果雅可比矩陣滿秩，可滿足要求。 (2)只有q=q*一個(gè)解決辦法 (3) 在1個(gè)3-2-1定位計(jì)劃中，一個(gè)約束方程的雅可比矩陣的滿秩的約束狀態(tài)如表1所示。如果定位是小于6，工件是限制約束的，即存在至少有一個(gè)工件自由定位議案不受限制的。如果矩陣滿秩，但定位大于6定位，工件是過約束，這表明存在至少一個(gè)定位等;而幾何約束工件被刪除不影響的狀態(tài)。找出一個(gè)模型除了3-2-1,可以建立基準(zhǔn)框架提取等效的定位點(diǎn)。胡等[6]已經(jīng)發(fā)展出一種系統(tǒng)的方法,對(duì)這個(gè)用途。因此,這則能適用于所有的定位方案。 圖1 .夾具系統(tǒng)模型。 表1 等級(jí) 數(shù)量的定位 地位 <6 - Under-constrained =6 =6 Well-constrained =6 >6 Over-constrained 康等[7]遵循這些方法和他們實(shí)施制定的幾何約束分析模塊其自動(dòng)化的計(jì)算機(jī)輔助夾具設(shè)計(jì)的核查制度。他們的CAFDV系統(tǒng)可以計(jì)算出雅可比矩陣和它的排名來確定定位的完整性。它也可以分析工件的位移和靈敏度定位錯(cuò)誤。熊等人[8]提出的等級(jí)檢查方法的定位矩陣WL(見附件)。他們還介紹了左/右邊的定位矩陣廣義逆理論,分析了工件的幾何誤差。結(jié)果表明,定位及發(fā)展方向誤差ΔX和位置誤差Δr的工件定位相關(guān)如下: Under-constrained:ΔX=WLΔr, (4) Well-constrained :ΔX=(WTLWL)-1WLTΔr, (5) Over-constrained:ΔX=WLT(WTLWL)-1Δr+(I6*6-WLT(WTLWL)-1WL)λ, (6) λ是任意一個(gè)向量。 他們還介紹了從這些矩陣的幾個(gè)指標(biāo)，評(píng)價(jià)定位配置，其次是通過約束非線性規(guī)劃的優(yōu)化。然而，他們的研究分析，不涉及非確定性定位的修訂。目前，還沒有就如何處理與提供確定的位置的夾具設(shè)計(jì)系統(tǒng)的研究。 2.定位完整性評(píng)價(jià) 如果不確定性的位置達(dá)到夾具系統(tǒng)設(shè)計(jì)的要求，設(shè)計(jì)師知道約束狀態(tài)是什么，以如何改善設(shè)計(jì)是非常重要的條件。如果夾具系統(tǒng)是過度約束，是理想定位需要的不必要的信息。而下約束時(shí)，所有有關(guān)知識(shí)約束工件的議案，可以引導(dǎo)設(shè)計(jì)師選擇額外的定位或使得修改定位計(jì)劃更有效。的總體戰(zhàn)略定位計(jì)劃表征幾何約束的狀態(tài)描述圖 2。 在本文中,定位矩陣秩的幾何約束的施加評(píng)價(jià)狀態(tài)(見附件為獲得的定位矩陣)。確定需要六個(gè)定位器定位提供矩陣的滿秩定位WL: 如圖3所示,在給定的定位器數(shù)量n,定位法向量[ai,bi,ci]和定位的位置[xi,yi,zi] 每一個(gè)定位器,i=1,2,.....,n,n*6定位矩陣可以確定如下: （7） 當(dāng)?shù)燃?jí)（WL）=6，n=6時(shí)，是工件良好約束。 當(dāng)?shù)燃?jí)（WL）=6，n>6時(shí);是工件過約束。 這意味著（n-6）有不必要的定位在定位方案上。工件將不存在限制（n-6）定位器。這種狀態(tài)的數(shù)學(xué)表示方法,那就是（n-6）在定位向量矩陣,可表示為線性組合的其他六行向量。 圖2 幾何約束狀態(tài)描述 圖3一個(gè)簡化的定位方案。 定位方案，提供了確定性的位置。發(fā)達(dá)國家的算法使用下列方法確定不必要的定位： 1、找到所有的（n-6）組合定位的。 2、為每個(gè)組合,從（n-6）定位器確定定位方案。 3、重新計(jì)算矩陣秩的定位為左六個(gè)定位器。 4、如果等級(jí)不變,被刪除的(n-6)定位器是負(fù)責(zé)過約束狀態(tài)。 這種方法可能會(huì)產(chǎn)生多種解決方案，并要求設(shè)計(jì)師來決定哪一套不必要的定位應(yīng)該被刪除以最佳定位性能。 當(dāng)?shù)燃?jí)（WL）<6,工件的限制約束。 參考文獻(xiàn) [1] Asada H, By AB.。自動(dòng)重構(gòu)夾具的柔性裝配夾具的運(yùn)動(dòng)學(xué)分析。 IEEE J機(jī)器人autom1985; RA-1:86-93。 [2] zhou YC，Chandru V，Barash MM。加工裝置的自動(dòng)配置的數(shù)學(xué)方法分析和綜合。反ASME J英工業(yè)1989;111:299-306。 [3] Wang MY, Liu T, Pelinescu DM.。夾具運(yùn)動(dòng)學(xué)分析的基礎(chǔ)上充分接觸剛體模型。 J制造業(yè)│科學(xué)與工程2003;125：316-24。 [4] Carlson JS。剛性零件的裝夾和定位計(jì)劃的二次靈敏度分析“。 ASME J制造業(yè)│2001年科學(xué)與工程;123（3）：462-72。 [5] Marin R, Ferreira P.確定性3-2-1定位計(jì)劃的運(yùn)動(dòng)學(xué)分析和綜合加工裝置。 ASME J制造業(yè)│科學(xué)與工程2001年;123:708-19。 [6] Hu W.設(shè)置規(guī)劃和公差分析。博士論文中，伍斯特理工學(xué)院;2001年。 [7] Kang Y, Rong Y, Yang J, Ma W.計(jì)算機(jī)輔助夾具設(shè)計(jì)驗(yàn)證。大會(huì)Autom2002;22:350-9。 [8] Rong KY, Huang SH, Hou Z.先進(jìn)的計(jì)算機(jī)輔助夾具設(shè)計(jì)。波士頓：愛思唯爾;2005年。 軸套鉆孔夾具設(shè)計(jì) 目錄 軸套鉆孔夾具設(shè)計(jì) 1 1.1軸套工藝分析 2 1.2定位基準(zhǔn)與機(jī)床刀具的選擇 2 1.3定位元件的選擇 2 1.4定位誤差分析與計(jì)算 4 1.5鉆套的設(shè)計(jì) 5 1.6夾緊裝置的設(shè)計(jì) 7 1.7 夾具操作的簡要說明 7 本夾具主要用來鉆軸套上端面的4-φ11孔的鉆孔夾具，對(duì)該4-φ11孔進(jìn)行專用夾具設(shè)計(jì)。以下是對(duì)該夾具的定位及夾緊方式進(jìn)行分析。 1.1軸套工藝分析 該零件主要是中間孔的加工，以及包括端面的4-φ11孔，還有鍵槽和外圓面上的螺孔。主要加工工藝分析如下： 零件可以采用鍛打料制成，主要加工工藝流程如： 1、 鍛打成毛坯 2、 粗車外圓、臺(tái)階外圓及端面 3、 鉆中間孔φ35，先鉆底孔后擴(kuò)孔最后用鉸刀鉸孔至公差要求 4、 鉆端面上的4-φ11孔 5、 銑中間鍵槽 6、 鉆φ55外圓面上的M8孔 1.2定位基準(zhǔn)與機(jī)床刀具的選擇 根據(jù)需要加工本次的軸套端面上的4-φ11孔的方式，本次采用搖臂鉆床上加工。搖臂鉆床的型號(hào)為Z3050，鉆頭采用φ11的麻花鉆。 本次鉆4-φ11孔的定位基準(zhǔn)按照工件的特征，選擇工件的φ55外圓為主要定位基準(zhǔn)，φ110端面下方采用輔助定位基準(zhǔn)。工件以φ55外圓定位用V型塊作為主要定位元件，其限制工件的X方向和Z方向的四個(gè)自由度，另外外圓φ110外圓面選用限制銷釘限制其另外Y方向的兩個(gè)自由度，這樣工件便被限制了六個(gè)自由度。 1.3定位元件的選擇 本次選擇的定位元件均為機(jī)床夾具的標(biāo)準(zhǔn)元件，定位V型塊為標(biāo)準(zhǔn)件，標(biāo)準(zhǔn)號(hào)為JB/T8018.1-1999，限制Z方向的支撐釘?shù)臉?biāo)準(zhǔn)號(hào)為JB/T8029.1-1999，φ110外圓面的擋銷為自制機(jī)加件。此定位元件圖形分別如下： 圖1-1 V型塊的標(biāo)準(zhǔn)件 圖1-2 支撐釘?shù)臉?biāo)準(zhǔn)件 1.4定位誤差分析與計(jì)算 根據(jù)本次的軸套的鉆4-φ11孔得出的： （1）工序基準(zhǔn) 在工序圖上用來確定加工表面的位置所依據(jù)的基準(zhǔn)。工序基準(zhǔn)可簡單地理解為工序圖上的設(shè)計(jì)基準(zhǔn)。分析計(jì)算定位誤差時(shí)所提到的設(shè)計(jì)基準(zhǔn)，是指零件圖上的設(shè)計(jì)基準(zhǔn)或工序圖上的工序基準(zhǔn)。本次的軸套的工序基準(zhǔn)為φ35的中心孔基準(zhǔn)。 （2）定位基準(zhǔn) 在加工過程中使工件占據(jù)正確加工位置所依據(jù)的基準(zhǔn)，即為工件與夾具定位元件定位工作面接觸或配合的表面。為提高工件的加工精度，應(yīng)盡量選設(shè)計(jì)基準(zhǔn)作定位基準(zhǔn)。本次的定位基準(zhǔn)為外圓φ55面。 如定位分析圖，對(duì)刀基準(zhǔn)是工件的A面，定位基準(zhǔn)為工件的φ55外圓，則基準(zhǔn)位置誤差為圖中O1點(diǎn)到O2點(diǎn)的距離。在ΔO1CO2中，，根據(jù)勾股定理求得 ①圖中尺寸的定位誤差 需要說明的是尺寸定位誤差的合成問題。由于和中都含有，即外圓直徑的變化同時(shí)引起和的變化，因而要判別二者合成時(shí)的符號(hào)。當(dāng)外圓直徑由大變小時(shí)，設(shè)計(jì)基準(zhǔn)相對(duì)定位基準(zhǔn)向上偏移，而當(dāng)此圓放入V形塊中定位時(shí)，因外圓直徑的變小，定位基準(zhǔn)相對(duì)調(diào)刀基準(zhǔn)是向下偏移的，二者變動(dòng)方向相反。故設(shè)計(jì)基準(zhǔn)相對(duì)對(duì)刀基準(zhǔn)的位移是二者之差，即 將68mm和90°代入，運(yùn)用V型塊定位，取α=90°，這時(shí)的定位誤差，定位誤差=Td/（2sin1/α）-Td/2=0.023 0.023×3＝0.069＜0.08 滿足要求。 1.5鉆套的設(shè)計(jì) 夾具在機(jī)床上安裝完畢，在進(jìn)行加工的時(shí)候，由于本次是鉆孔，需要用導(dǎo)向件使得鉆套下鉆可以增強(qiáng)同軸度的特性。本次選用的鉆套采用可換鉆套，連同鉆套并選用鉆套螺釘將鉆套固定，選用的可換鉆套和鉆套螺釘均為機(jī)床夾具的標(biāo)準(zhǔn)件，本次設(shè)計(jì)采用可換鉆套，國標(biāo)號(hào)為JB/T8045.2-1999，鉆套螺釘?shù)膰鴺?biāo)號(hào)為JB/T8045.5-1999。本次設(shè)計(jì)采用的鉆套和鉆套螺釘?shù)膱D形如下： 圖1-4鉆套螺釘?shù)臉?biāo)準(zhǔn)件 圖1-5可換鉆套的標(biāo)準(zhǔn)件 1.6夾緊裝置的設(shè)計(jì) 本次的夾緊采用外圓的夾緊方式，原因是以工件的外圓φ55外圓為定位，采用的是V型塊，即另一邊采用活動(dòng)V型塊對(duì)工件進(jìn)行夾緊，此夾緊的效果同三爪卡盤的夾緊方式相同。夾緊的裝置如下圖所示： 圖1-5夾緊裝置設(shè)計(jì) 1.7 夾具操作的簡要說明 本次夾具的操作較為簡單，只需轉(zhuǎn)動(dòng)側(cè)邊的固定手柄壓緊螺釘上的轉(zhuǎn)動(dòng)手柄即可。 1.8 參考文獻(xiàn) [1] 王先逵編著．機(jī)械制造工藝學(xué)（上下冊(cè)）[M]．北京：清華大學(xué)出版社，1989 ． [2] 李德慶，吳錫英編著．計(jì)算機(jī)輔助制造[M]．北京：清華大學(xué)出版社，1992 ． [3] 趙志修主著．機(jī)械制造工藝學(xué)[M]．北京：機(jī)械工業(yè)出版社，1985 ． [4] 孫大涌主編．先進(jìn)制造技術(shù)[M]．北京：機(jī)械工業(yè)出版社，2000 ． 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