夾具類外文翻譯-采用遺傳算法優(yōu)化加工夾具定位和加緊位置【中文4477字】【PDF+中文WORD】,中文4477字,PDF+中文WORD,夾具,外文,翻譯,采用,遺傳,算法,優(yōu)化,加工,定位,加緊,位置,中文,4477,PDF,WORD 采用遺傳算法優(yōu)化加工夾具定位和加緊位置 摘要：工件變形的問題可能導致機械加工中的空間問題。支撐和定位器是用于減少工件彈性變形引起的誤差。支撐、定位器的優(yōu)化和夾具定位是最大限度的減少幾何在工件加工中的誤差的一個關鍵問題。本文應用夾具布局優(yōu)化遺傳算法（GAs）來處理夾具布局優(yōu)化問題。遺傳算法的方法是基于一種通過整合有限的運行于批處理模式的每一代的目標函數(shù)值的元素代碼的方法，用于來優(yōu)化夾具布局。給出的個案研究說明已開發(fā)的方法的應用。采用染色體文庫方法減少整體解決問題的時間。已開發(fā)的遺傳算法保持跟蹤先前的分析設計，因此先前的分析功能評價的數(shù)量降低大約93%。結果表明，該方法的夾具布局優(yōu)化問題是多模式的問題。優(yōu)化設計之間沒有任何明顯的相似之處，雖然它們提供非常相似的表現(xiàn)。 關鍵詞：夾具設計；遺傳算法；優(yōu)化 1.引言 夾具用來定位和束縛機械操作中的工件，減少由于對確保機械操作準確性的夾緊方案和切削力造成的工件和夾具的變形。傳統(tǒng)上，加工夾具是通過反復試驗法來設計和制造的，這是一個既造價高又耗時的制造過程。為確保工件按規(guī)定尺寸和公差來制造，工件必須給予適當?shù)亩ㄎ缓蛫A緊以確保有必要開發(fā)工具來消除高造價和耗時的反復試驗設計方法。適當?shù)墓ぜㄎ缓蛫A具設計對于產(chǎn)品質(zhì)量的精密度、準確度和機制件的完飾是至關重要的。 從理論上說，3-2-1定位原則對于定位所有的棱柱形零件是很令人滿意的。該方法具有最大的剛性與最少量的夾具元件。從動力學觀點來看定位零件意味著限制了自由移動物體的六自由度（三個平動自由度和三個旋轉(zhuǎn)自由度）。在零件下部設置三個支撐來建立工件在垂直軸方向的定位。在兩個外圍邊緣放置定位器旨在建立工件在水平x軸和y軸的定位。正確定位夾具的工件對于制造過程的全面準確性和重復性是至關重要的。定位器應該盡可能的遠距離的分開放置并且應該放在任何可能的加工面上。放置的支撐器通常用來包圍工件的重力中心并且盡可能的將其分開放置以維持其穩(wěn)定性。夾具夾子的首要任務是固定夾具以抵抗定位器和支撐器。不應該要求夾子反抗加工操作中的切削力。 對于給定數(shù)量的夾具元件，加工夾具合成的問題是尋找夾具優(yōu)化布局或工件周圍夾具元件的位置。本篇文章提出一種優(yōu)化夾具布局遺傳算法。優(yōu)化目標是研究一個二維夾具布局使工件不同位置上最大的彈性變形最小化。ANSYS程序以用于計算工件變形情況下夾緊力和切削力。本文給出兩個實例來說明給出的方法。 2.回顧相關工程結構 最近幾年夾具設計問題受到越來越多的重視。然而，很少有注意力集中于優(yōu)化夾具布局設計。Menassa和Devries用FEA計算變形量使設計準則要求的位點的工件變形最小化。設計問題是確定支撐器位置。Meyer和Liou提出一個方法就是使用線性編程技術合成動態(tài)編程條件中的夾具。給出了使夾緊力和定位力最小化的解決方案。Li和Melkote用非線性規(guī)劃方法解決布局優(yōu)化問題。這個方法使工件位置誤差最小化歸于工件的局部彈性變形。Roy和Liao開發(fā)出一種啟發(fā)式方法來計劃最好的支撐和夾緊位置。Tao等人提出一個幾何推理的方法來確定最優(yōu)夾緊點和任意形狀工件的夾緊順序。Liao和Hu提出一種夾具結構分析系統(tǒng)這個系統(tǒng)基于動態(tài)模型分析受限于時變加工負載的夾具—工件系統(tǒng)。本文也調(diào)查了夾緊位置的影響。Li和Melkote提出夾具布局和夾緊力最優(yōu)合成方法幫我們解釋加工過程中的工件動力學。本文提出一個夾具布局和夾緊力優(yōu)化結合的程序。他們用接觸彈性建模方法解釋工件剛體動力學在加工期間的影響。Amaral等人用ANSYS驗證夾具設計的完整性。他們用3-2-1方法。ANSYS提出優(yōu)化分析。Tan等人通過力鎖合、優(yōu)化與有限建模方法描述了建模、優(yōu)化夾具的分析與驗證。 以上大部分的研究使用線性和非線性編程方式這通常不會給出全局最優(yōu)解決方案。所有的夾具布局優(yōu)化程序開始于一個初始可行布局。這些方法給出的解決方案在很大程度上取決于初始夾具布局。他們沒有考慮到工件夾具布局優(yōu)化對整體的變形。 GAs已被證明在解決工程中優(yōu)化問題是有用的。夾具設計具有巨大的解決空間并需要搜索工具找到最好的設計。一些研究人員曾使用GAs解決夾具設計及夾具布局問題。Kumar等人用GAs和神經(jīng)網(wǎng)絡設計夾具。Marcelin已經(jīng)將GAs用于支撐位置的優(yōu)化。Vallapuzha等人提出基于優(yōu)化方法的GA，它采用空間坐標來表示夾具元件的位置。夾具布局優(yōu)化程序設計的實現(xiàn)是使用MATLAB和遺傳算法工具箱。HYPERMESH和MSC / NASTRAN用于FE模型。Vallapuzha等人提出一些結果關于一個廣泛調(diào)查不同優(yōu)化方法的相對有效性。他們的研究表明連續(xù)遺傳算法提出了最優(yōu)質(zhì)的解決方案。Li和Shiu使用遺傳算法確定了夾具設計最優(yōu)配置的金屬片。MSC/NASTRAN已經(jīng)用于適應度值評價。Liao提出自動選擇最佳夾子和夾鉗的數(shù)目以及它們在金屬片整合的夾具中的最優(yōu)位置。Krishnakumar和Melkote開發(fā)了一種夾具布局優(yōu)化技術，它是利用遺傳算法找到了夾具布局，由于整個刀具路徑中的夾緊力和加工力使加工表面變形量最小化。通過節(jié)點編號使定位器和夾具位置特殊化。一個內(nèi)置的有限元求解器研制成功。 一些研究沒考慮到整個刀具路徑的優(yōu)化布局以及磨屑清除。一些研究采用節(jié)點編號作為設計參數(shù)。 在本研究中，開發(fā)GA工具用于尋找在二維工件中的最優(yōu)定位器和夾緊位置。使用參考邊緣的距離作為設計參數(shù)而不是用FEA節(jié)點編號。真正編碼遺傳算法的染色體的健康指數(shù)是從FEA結果中獲得的。ANSSYS用于FEA計算。用染色體文庫的方法是為了減少解決問題的時間。用兩個問題測試已開發(fā)的遺傳算法工具。給出的兩個實例說明了這個開發(fā)的方法。本論文的主要貢獻可以概括為以下幾個方面： （1） 開發(fā)了遺傳算法編碼結合商業(yè)有限元素求解； （2） 遺傳算法采用染色體文庫以降低計算時間； （3） 使用真正的設計參數(shù)，而不是有限元節(jié)點數(shù)字； （4） 當工具在工件中移動時考慮磨屑清除工具。 3.遺傳算法概念 遺傳算法最初由John Holland開發(fā)。Goldberg出版了一本書，解釋了這個理論和遺傳算法應用實例的詳細說明。遺傳算法是一種隨機搜索方法，它模擬一些自然演化的機制。該算法用于種群設計。種群從一代到另一代演化，通過自然選擇逐漸提高了適應環(huán)境的能力，更健康的個體有更好的機會，將他們的特征傳給后代。 該算法中，要基于為每個設計計算適合性，所以人工選擇取代自然環(huán)境選擇。適應度值這個詞用來指明染色體生存幾率，它在本質(zhì)上是該優(yōu)化問題的目標函數(shù)。生物定義的特征染色體用代表設計變量的字符串中的數(shù)值代替。 被公認的遺傳算法與傳統(tǒng)的梯度基礎優(yōu)化技術的不同主要有如下四種方式： （1） 遺傳算法和問題中的一種編碼的設計變量和參數(shù)一起工作而不是實際參數(shù)本身。 （2） 遺傳算法使用種群—類型研究。評價在每個重復中的許多不同的設計要點而不是一個點順序移動到下一個。 （3） 遺傳算法僅僅需要一個適當?shù)幕蚰繕撕瘮?shù)值。沒有衍生品或梯度是必要的。 （4） 遺傳算法以用概率轉(zhuǎn)換規(guī)則來發(fā)現(xiàn)新設計為探索點而不是利用基于梯度信息的確定性規(guī)則來找到這些新觀點。 4.方法 4.1夾具定位原則 加工過程中，用夾具來保持工件處于一個穩(wěn)定的操作位置。對于夾具最重要的標準是工件位置精確度和工件變形。一個良好的夾具設計使工件幾何和加工精度誤差最小化。另一個夾具設計的要求是夾具必須限制工件的變形?？紤]切削力以及夾緊力是很重要的。沒有足夠的夾具支撐，加工操作就不符合設計公差。有限元分析在解決這其中的一些問題時是一種很有力的工具。 棱柱形零件常見的定位方法是3-2-1方法。該方法具有最大剛體度以及最小夾具元件數(shù)。在三維中一個工件可能會通過六自由度定位方法快速定位為了限制工件的九個自由度。其他的三個自由度通過夾具元件消除了?；?-2-1定位原理的二位工件布局的例子如圖4。 圖4 3-2-1對二維棱柱工件定位布局 定位面得數(shù)量不得超過兩個避免冗余的位置?；?-2-1的夾具設計原則有兩種精確的定位平面包含于兩個或一個定位器。因此，在兩邊有最大的夾緊力抵抗每個定位平面。夾緊力總是指向定位器為了推動工件接觸到所有的定位器。定位點對面應定位夾緊點防止工件由于夾緊力而扭曲。因為加工力沿著加工面，所以有必要確保定位器的反應力在所有時間內(nèi)是正的。任何負面的反應力表示工件從夾具元件中脫離。換句話說，當反應力是負的時候，工件和夾具元件之間接觸或分離的損失可能發(fā)生。定位器內(nèi)正的反應力確保工件從切削開始到結束都能接觸到所有的定位器。夾緊力應該充分束縛和定位工件且不導致工件的變形或損壞。本文不考慮夾緊力的優(yōu)化。 4.2基于夾具布局優(yōu)化方法的遺傳算法 在實際設計問題中，設計參數(shù)的數(shù)量可能很大并且它們對目標函數(shù)的影響會是非常復雜的。目標函數(shù)曲線必須是光滑的并且需要一個程序計算梯度。遺傳算法在理念上遠不同于其他的探究方法，它們包括傳統(tǒng)的優(yōu)化方法和其他隨機方法。通過運用遺傳算法來對夾具優(yōu)化布局，可以獲得一個或一組最優(yōu)的解決方案。 本項研究中，最優(yōu)定位器和夾具定位使用遺傳算法確定。它們是理想的適合夾具布局優(yōu)化問題的方法因為沒有直接分析的關系存在于加工誤差和夾具布局中。因為遺傳算法僅僅為一個特別的夾具布局處理設計變量和目標函數(shù)值，所以不需要梯度或輔助信息。 建議方案流程圖如圖5。 使用開發(fā)的命名為GenFix的Delphi語言軟件來實現(xiàn)夾具布局優(yōu)化。位移量用ANSYS軟件計算。通過WinExec功能在GenFix中運行ANSYS很簡單。GenFix和ANSYS之間相互作用通過四部實現(xiàn)： （1） 定位器和夾具位置從二進制代碼字符串中提取作為真正的參數(shù)。 （2） 這些參數(shù)和ANSYS輸入批處理文件（建模、解決方案和后置處理）用WinExec功能傳給ANSYS。 （3） 解決后將位移值寫成一個文本文件。 （4） GenFix讀這個文件并為當前定位器和夾緊位置計算適應度值。 為了減少計算量，染色體與適應度值儲存在一個文庫里以備進一步評估。GenFix首先檢查是否當前的染色體的適應度值已經(jīng)在之前被計算過。如果沒有，定位器位置被送到ANSYS，否則從文庫中取走適應度值。在初始種群產(chǎn)生過程中，檢查每一個染色體可行與否。如果違反了這個原則，它就會出局然后新的染色體就產(chǎn)生了。這個程序創(chuàng)造了可行的初始種群。這保證了初始種群的每個染色體在夾緊力和切削力作用下工件的穩(wěn)定性。用兩個測試用例來驗證提到的遺傳算法計劃。第一個實例是使用Himmelblau功能。在第二個測試用例中，遺傳算法計劃用來優(yōu)化均布載荷作用下梁的支撐位置。 圖5 設計方法的流程與ANSYS相配合流程 5.夾具布局優(yōu)化的個案研究 該夾具布局優(yōu)化問題的定義是：找到定位器和夾子的位置以使在特定區(qū)工件變形降到最小程度。那么多的定位器和夾子并不是設計參數(shù)因為它們在3-2-1方案中是已知的和固定的。因此，設計參數(shù)的選擇如同定位器和夾子的位置。本研究中不考慮摩擦力。兩個實例研究來說明以提出的方法。 6.結論 本文提出了一個夾具布局優(yōu)化的評價優(yōu)化技術。ANSYS用于FE計算適應度值?？梢钥吹?，遺傳算法和FE方法的結合對當今此類問題似乎是一種強大的方法。遺傳算法特別適合應用于解決那些在目標函數(shù)和設計變量之間不存在一個定義明確的數(shù)學關系的問題。結果證明遺傳算法在夾具布局優(yōu)化問題方面的成功應用。本項研究中，遺傳算法在夾具布局優(yōu)化應用中的主要困難是較高的計算成本。種群中每個染色體需要工件的重嚙合。但是，染色體庫的使用，F(xiàn)E評價的數(shù)量從6000下降到415。這就導致了巨大的增益計算效益。其他減少處理時間的方法是在局域網(wǎng)內(nèi)使用分布式計算。 該方法結果表明，夾具布局優(yōu)化問題是多模態(tài)問題。優(yōu)化設計之間沒有任何明顯的相似之處盡管他們提供非常相似的表現(xiàn)。結果表明夾具布局問題是多模態(tài)問題然而用于夾具設計的啟發(fā)式規(guī)則應該用于遺傳算法來選擇最優(yōu)的設計。 Machining fixture locating and clamping position optimizationusing genetic algorithmsNecmettin Kaya*Department of Mechanical Engineering,Uludag University,Go ru kle,Bursa 16059,TurkeyReceived 8 July 2004;accepted 26 May 2005Available online 6 September 2005AbstractDeformationoftheworkpiecemaycausedimensionalproblemsinmachining.Supportsandlocatorsareusedinordertoreducetheerrorcausedby elastic deformation of the workpiece.The optimization of support,locator and clamp locations is a critical problem to minimize the geometricerror in workpiece machining.In this paper,the application of genetic algorithms(GAs)to the fixture layout optimization is presented to handlefixture layout optimization problem.A genetic algorithm based approach is developed to optimise fixture layout through integrating a finiteelement code running in batch mode to compute the objective function values for each generation.Case studies are given to illustrate theapplicationofproposedapproach.Chromosomelibraryapproachisusedtodecreasethetotalsolutiontime.DevelopedGAkeepstrackofprevioslyanalyzed designs,therefore the number of function evaulations are decreased about 93%.The results of this approach show that the fixture layoutoptimization problems are multi-modal problems.Optimized designs do not have any apparent similarities although they provide very similarperformances.#2005 Elsevier B.V.All rights reserved.Keywords:Fixture design;Genetic algorithms;Optimization1.IntroductionFixtures are used to locate and constrain a workpiece duringa machining operation,minimizing workpiece and fixturetooling deflections due to clamping and cutting forces arecritical to ensuring accuracy of the machining operation.Traditionally,machining fixtures are designed and manufac-tured through trial-and-error,which prove to be both expensiveand time-consuming to the manufacturing process.To ensure aworkpiece is manufactured according to specified dimensionsand tolerances,it must be appropriately located and clamped,making it imperative to develop tools that will eliminate costlyand time-consuming trial-and-error designs.Proper workpiecelocation and fixture design are crucial to product quality interms of precision,accuracy and finish of the machined part.Theoretically,the 3-2-1 locating principle can satisfactorilylocate all prismatic shaped workpieces.This method providesthe maximum rigidity with the minimum number of fixtureelements.To position a part from a kinematic point of viewmeans constraining the six degrees of freedom of a free movingbody(three translations and three rotations).Three supports arepositioned below the part to establish the location of theworkpiece on its vertical axis.Locators are placed on twoperipheral edges and intended to establish the location of theworkpiece on the x and y horizontal axes.Properly locating theworkpiece in the fixture is vital to the overall accuracy andrepeatability of the manufacturing process.Locators should bepositioned as far apart as possible and should be placed onmachined surfaces wherever possible.Supports are usuallyplaced to encompass the center of gravity of a workpiece andpositioned as far apart as possible to maintain its stability.Theprimary responsibility of a clamp in fixture is to secure the partagainstthelocatorsandsupports.Clampsshouldnotbeexpectedto resist the cutting forces generated in the machining operation.For a given number of fixture elements,the machiningfixture synthesis problem is the finding optimal layout orpositions of the fixture elements around the workpiece.In thispaper,a method for fixture layout optimization using geneticalgorithms is presented.The optimization objective is to searchfor a 2D fixture layout that minimizes the maximum elasticdeformation at different locations of the workpiece.ANSYSprogram has been used for calculating the deflection of the in Industry 57(2006)112120*Tel.:+90 224 4428176;fax:+90 224 4428021.E-mail address:necmiuludag.edu.tr.0166-3615/$see front matter#2005 Elsevier B.V.All rights reserved.doi:10.1016/pind.2005.05.001under clamping and cutting forces.Two case studies are givento illustrate the proposed approach.2.Review of related worksFixture design has received considerable attention in recentyears.However,little attention has been focused on theoptimum fixture layout design.Menassa and DeVries 1 usedFEA for calculating deflections using the minimization of theworkpiece deflection at selected points as the design criterion.The design problem was to determine the position of supports.Meyer and Liou 2 presented an approach that uses linearprogramming technique to synthesize fixtures for dynamicmachining conditions.Solution for the minimum clampingforces and locator forces is given.Li and Melkote 3 used anonlinear programming method to solve the layout optimiza-tion problem.The method minimizes workpiece location errorsdue to localized elastic deformation of the workpiece.Roy andLiao 4 developed a heuristic method to plan for the bestsupporting and clamping positions.Tao et al.5 presented ageometricalreasoning methodologyfor determining theoptimal clamping points and clamping sequence for arbitrarilyshaped workpieces.Liao and Hu 6 presented a system forfixture configuration analysis based on a dynamic model whichanalyses the fixtureworkpiece system subject to time-varyingmachining loads.The influence of clamping placement is alsoinvestigated.Li and Melkote 7 presented a fixture layout andclamping force optimal synthesis approach that accounts forworkpiece dynamics during machining.A combined fixturelayout and clamping force optimization procedure presented.They used the contact elasticity modeling method that accountsfor the influence of workpiece rigid body dynamics duringmachining.Amaral et al.8 used ANSYS to verify fixturedesign integrity.They employed 3-2-1 method.The optimiza-tion analysis is performed in ANSYS.Tan et al.9 describedthe modeling,analysis and verification of optimal fixturingconfigurations by the methods of force closure,optimizationand finite element modeling.Mostoftheabovestudiesuselinearornonlinearprogramming methods which often do not giveglobal optimumsolution.All of the fixture layout optimization procedures startwith an initial feasible layout.Solutionsfrom these methods aredepend on the initial fixture layout.They do not consider thefixture layout optimization on overall workpiece deformation.The GAs have been proven to be useful technique in solvingoptimization problems in engineering 1012.Fixture designhas a large solution space and requires a search tool to find thebest design.Few researchers have used the GAs for fixturedesign and fixture layout problems.Kumar et al.13 haveapplied both GAs and neural networks for designing a fixture.Marcelin 14 has used GAs to the optimization of supportpositions.Vallapuzhaetal.15presentedGAbasedoptimization method that uses spatial coordinates to representthe locations of fixture elements.Fixture layout optimizationprocedure was implemented using MATLAB and the geneticalgorithm toolbox.HYPERMESH and MSC/NASTRAN wereusedforFEmodel.Vallapuzhaetal.16 presentedresults ofanextensive investigation into the relative effectiveness of variousoptimization methods.They showed that continuous GAyielded the best quality solutions.Li and Shiu 17 determinedthe optimal fixture configuration design for sheet metalassembly using GA.MSC/NASTRAN has been used forfitness evaulation.Liao 18 presented a method to auto-matically select the optimal numbers of locators and clamps aswell as their optimal positions in sheet metal assembly fixtures.Krishnakumar and Melkote 19 developed a fixture layoutoptimization technique that uses the GA to find the fixturelayout that minimizes the deformation of the machined surfacedue to clamping and machining forces over the entire tool path.Locator and clamp positions specified by node numbers.Abuilt-in finite element solver was developed.Some of the studies do not consider the optimization of thelayout for entire tool path and chip removal is not taken intoaccount.Some of the studies used node numbers as designparameters.In this study,a GA tool has been developed to find theoptimal locator and clamp positions in 2D workpiece.Distances from the reference edges as design parameters areused rather than FEA node numbers.Fitness values of realencoded GA chromosomes are obtained from the results ofFEA.ANSYS has been used for FEA calculations.Achromosome library approach is used in order to decreasethe solution time.Developed GA tool is tested on two testproblems.Two case studies are givento illustrate the developedapproach.Main contributions of this paper can be summarizedas follows:(1)developed a GA code integrated with a commercial finiteelement solver;(2)GA uses chromosome library in order to decrease thecomputation time;(3)real design parameters are used rather than FEA nodenumbers;(4)chip removal is taken into account while tool forces movingon the workpiece.3.Genetic algorithm conceptsGenetic algorithms were first developed by John Holland.Goldberg 10 published a book explaining the theory andapplication examples of genetic algorithm in details.A geneticalgorithm is a random search technique that mimics somemechanisms of natural evolution.The algorithm works on apopulation of designs.The population evolves from generationto generation,gradually improving its adaptation to theenvironment through natural selection,fitter individuals havebetter chances of transmitting their characteristics to latergenerations.In the algorithm,the selection of the natural environment isreplaced by artificial selection based on a computed fitness foreach design.The term fitness is used to designate thechromosomes chances of survival and it is essentially theobjective function of the optimization problem.The chromo-somes that define characteristics of biological beings areN.Kaya/Computers in Industry 57(2006)112120113replaced by strings of numerical values representing the designvariables.GA is recognized to be different than traditional gradient-basedoptimizationtechniquesinthefollowingfour major ways10:1.GAs work with a coding of the design variables andparameters in the problem,rather than with the actualparameters themselves.2.GAs make use of population-type search.Many differentdesign points are evaluated during each iteration instead ofsequentially moving from one point to the next.3.GAs need only a fitness or objective function value.Noderivatives or gradients are necessary.4.GAs use probabilistic transition rules to find new designpoints for exploration rather than using deterministic rulesbased on gradient information to find these new points.Algorithm of the basic GA is given as follows:1.Initial population:Generate random population of chromo-somes.2.Fitness:Evaluate the fitness of each chromosome in thepopulation.3.Test:If the end condition is satisfied,stop,and return the bestsolution in current population.4.New population:Create a new population by repeatingfollowing steps until the new population is complete.Reproduction:Select two parent chromosomes from thepopulation according to their fitness.Crossover:With a crossover probability,crossover theparents to form a new offspring(children).If no crossoverwas performed,offspring is an exact copy of parents.Mutation:With a mutation probability,mutate new offspringat each locus(position in chromosome).5.Replace:Use new generated population for a further run ofalgorithm.6.Loop:Go to step 2.3.1.Individual representationThe first andmostimportantstep in preparing anoptimization problem for a GA solution is that of defining aparticular coding of the design variables and their arrangementinto a string of numerical values to be used as the chromosomeby the GA.In most GAs,finite length binary coded strings of ones andzeros are used to describe the parameters for each solution.In amultiparameter optimization problem,individual parametercoding are usually concatenated into a complete string which isshown in Fig.1.In this paper,real representation of binary string is used.Thelength of the string depends on the required precision.Themapping from a binary string to a real number is completed intwo steps:Step 1:Find code length for xi(i=1,.,n):c xmaxi?xmini?rwhere r is the required precision(101,102,103,.).Code length for xiis as follows:lxi n 1where,2nc2n1Total string length is given by:l Xni1lxiStep 2:Mapping from a binary string to a real number:xi xminixmaxi?xmini2n?1Xnj1qij2j?1where qij2 0,1.In order to generate the chromosomes,the length of thechromosome is calculated first.Then random numbers in therange of 0,1aregenerated toform the chromosome.Randomfunction is used in Delphi programming language as a randomnumber generator.3.2.Genetic operatorsEstablishing the GA parameters is very crucial in anoptimization problem because there are no guidelines 20.Thegenetic algorithms contains several operators,e.g.reproduc-tion,crossover,mutation,etc.3.2.1.ReproductionThe reproduction operator allows individual strings to becopied for possible inclusion in the next generation.Afterassesingthefitness valuefor eachstringinthe initialpopulation,only a few strings with high fitness value are considered in thereproduction.There are many different types of reproductionoperatorswhichareproportionalselection,tournamentselection,ranking selection,etc.In this study,tournament selection isselected,since it has better convergence and computational timecomparedtoanyotherreproductionoperator11.Intournamentselection,two individuals are choosen from the population atrandom.Then the string which has best fitness value is selected.This procedure is continued until the size of the reproductionpopulation is equal to the size of the population.3.2.2.CrossoverCrossoveristhenextoperationinthegeneticalgorithm.Thisoperation partially exchanges information between any twoN.Kaya/Computers in Industry 57(2006)112120114Fig.1.Binary representation in GA.selected individuals.Crossover selects genes from parentchromosomes and creates new offsprings.Like reproductionoperator,thereexistanumberofcrossoveroperatorsinGA.Inasingle-point crossoveroperator which is used in this paper,bothstrings are cut at an arbitrary place and the right-side portion ofboth strings are swapped among themselves to create two newstrings,as illustrated in Fig.2.In order to carry out the crossover operation,two individualsare selected from the population at random.Then a randomnumber in the range of 0,1 is generated.If this randomnumber is less than the probability of crossover then theseindividuals are subjected to crossover,otherwise they arecopiedtonewpopulationastheyare.Alsothecrossoverpointisselected at random.Probability of crossover(Pc)is selectedgenerally between 0.6 and 0.9.3.2.3.MutationThis is the process of randomly modifying the string withsmall probability.Mutation operator changes 10 and viceversa with a small probability of mutation(Pm).The need formutation is to keep diversity in the population 11.This is toprevent falling all solutions in population into a local optimumof solved problem.Fig.3 illustrates the mutation operation atseventh bit position.In order to determine whether a chromoseme is to besubjectedtomutation,arandomnumberintherangeof0,1isgenerated.If this random number is less than the probability ofmutation,selected chromosome will be mutated.Probability ofmutation should be selected very low as a high mutation willdestroy fit chromosomes and degenerate the GA into a randomwalk.Pmshould be selected between 0.02 and 0.06 21.3.2.4.Constraint handlingIn most application of GAs to constrained optimizationproblems,the penalty function method has been used.In thisstudy a method proposed by Deb 12 is used.Although apenalty term is added to the objective function,this methoddiffers from conventional GA implementations.The methodproposes to use a tournament selection operator,where twosolutions are compared at a time and the following criteria arealways enforced:-Any feasible solution is preferred to any infeasible solution.-Among two feasible solutions,the one having better fitnessvalue is preferred.-Among two infeasible solutions,the one having smallerconstraint violation is preferred.3.2.5.Elitist strategyIn this strategy,some of the best individuals are copied intothe next generation without applying any genetic operators.Elitist strategy always clones the best individuals of the currentgeneration into the next generation.This guarantees that thebest found design is never lost in future generations.4.Approach4.1.Fixture positioning principlesIn machining process,fixtures are used to keep workpiecesin a desirable position for operations.The most importantcriteria for fixturing are workpiece position accuracy andworkpiece deformation.A good fixture design minimizesworkpiece geometric and machining accuracy errors.Anotherfixturing requirement is that the fixture must limit deformationoftheworkpiece.Itisimportanttoconsiderthecuttingforcesaswell as the clamping forces.Without adequate fixture support,machining operations do not conform to designed tolerances.Finite element analysis is a powerful tool in the resolution ofsome of these problems 22.Common locating method for prismatic parts is 3-2-1method.This method provides the maximum rigidity with theminimum number of fixture elements.Aworkpiece in 3D maybe positively located by means of six points positioned so thatthey restrict nine degrees of freedom of the workpiece.Theother three degrees offreedom are removed by clamp elements.An example layout for 2D workpiece based 3-2-1 locatingprinciple is shown in Fig.4.The number of locating faces must not exceed two so as toavoid a redundant location.Based on the 3-2-1 fixturingprinciple there are two locating planes for accurate locationcontainingtwoand onelocators.Therefore,thereare maximumof two side clampings against each locating plane.Clampingforces are always directed towards the locators in order to forcethe workpiece to contact all locators.The clamping pointN.Kaya/Computers in Industry 57(2006)112120115Fig.2.Illustration of crossover operator.Fig.3.Illustration of mutation operator.Fig.4.3-2-1 locating layout for 2D prismatic workpiece.should be positioned opposite the positioning points to preventthe workpiece from being distorted by the clamping force.Since the machining forces travel along the machining area,it is necessary to ensure that the reaction forces at locators arepositive for all the time.Any negative reaction force indicatesthat theworkpiece is free from fixture elements.In other words,loss of contact or the separation between the workpiece andfixture element might happen when the reaction force isnegative.Positive reaction forces at the locators ensure that theworkpiece maintains contact with all the locators from thebeginning of the cut to the end.The clamping forces should bejust sufficient to constrain and locate the workpiece withoutcausing distortion or damage to the workpiece.Clamping forceoptimization is not considered in this paper.4.2.Genetic algorithm based fixture layout optimizationapproachIn real design problems,the number of design parameterscan be very large and their influence on the objective functioncan be very complicated.The objective function must besmooth and a procedure is needed to compute gradients.Genetic algorithms strongly differ in conception from othersearch methods,including traditional optimization methodsand other stochastic methods 23.By applying GAs to fixturelayout optimization,an optimal or group of sub-optimalsolutions can be obtained.In this study,optimum locator and clamp positions aredetermined using genetic algorithms.They are ideally suitedfor the fixture layout optimization problem since no directanalytical relationship exist between the machining error andthe fixture layout.Since the GA deals with only the designvariables and objective function value for a particular fixturelayout,no gradient or auxiliary information is needed 19