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1Machining fixture locating and clamping position optimization using genetic algorithmsNecmettin KayaDepartment of Mechanical Engineering, Uludag University, Go¨ru¨kle, Bursa 16059, Turkey Received 8 July 2004; accepted 26 May 2005Available online 6 September 2005AbstractDeformation of the workpiece may cause dimensional problems in machining. Supports and locators are used in order to reduce the error caused by elastic deformation of the workpiece. The optimization of support, locator and clamp locations is a critical problem to minimize the geometric error in workpiece machining. In this paper, the application of genetic algorithms (GAs) to the fixture layout optimization is presented to handle fixture layout optimization problem. A genetic algorithm based approach is developed to optimise fixture layout through integrating a finite element code running in batch mode to compute the objective function values for each generation. Case studies are given to illustrate the application of proposed approach. Chromosome library approach is used to decrease the total solution time. Developed GA keeps track of previously analyzed designs; therefore the numbers of function evaluations are decreased about 93%. The results of this approach show that the fixture layout optimization problems are multi-modal problems. Optimized designs do not have any apparent similarities although they provide very similar performances.Keywords: Fixture design; Genetic algorithms; Optimization1. IntroductionFixtures are used to locate and constrain a workpiece during a machining operation, minimizing workpiece and fixture tooling deflections due to clamping and cutting forces are critical to ensuring accuracy of the machining operation. Traditionally, machining fixtures are designed and manufactured through trial-and-error, which prove to be both expensive and time-consuming to the manufacturing process. To ensure a workpiece is manufactured according to specified dimensions and tolerances, it must be appropriately located and clamped, making it imperative to develop tools that will eliminate costly and time-consuming trial-and-error designs. Proper workpiece location and fixture design are crucial to product quality in terms of precision, accuracy and finish of the machined part. Theoretically, the 3-2-1 locating principle can satisfactorily locate all prismatic shaped workpieces. This method provides the maximum rigidity with the minimum number of fixture 2elements. To position a part from a kinematic point of view means constraining the six degrees of freedom of a free moving body (three translations and three rotations). Three supports are positioned below the part to establish the location of the workpiece on its vertical axis. Locators are placed on two peripheral edges and intended to establish the location of the workpiece on the x and y horizontal axes. Properly locating the workpiece in the fixture is vital to the overall accuracy and repeatability of the manufacturing process. Locators should be positioned as far apart as possible and should be placed on machined surfaces wherever possible. Supports are usually placed to encompass the center of gravity of a workpiece and positioned as far apart as possible to maintain its stability. The primary responsibility of a clamp in fixture is to secure the part against the locators and supports. Clamps should not be expected to resist the cutting forces generated in the machining operation. For a given number of fixture elements, the machining fixture synthesis problem is the finding optimal layout or positions of the fixture elements around the workpiece. In this paper, a method for fixture layout optimization using genetic algorithms is presented. The optimization objective is to search for a 2D fixture layout that minimizes the maximum elastic deformation at different locations of the workpiece. ANSYS program has been used for calculating the deflection of the part under clamping and cutting forces. Two case studies are given to illustrate the proposed approach.2. Review of related worksFixture design has received considerable attention in recent years. However, little attention has been focused on the optimum fixture layout design. Menassa and DeVries[1]used FEA for calculating deflections using the minimization of the workpiece 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 linear programming technique to synthesize fixtures for dynamic machining conditions. Solution for the minimum clamping forces and locator forces is given. Li and Melkote[3]used a nonlinear programming method to solve the layout optimization problem. The method minimizes workpiece location errors due to localized elastic deformation of the workpiece. Roy andLiao[4]developed a heuristic method to plan for the best supporting and clamping positions. Tao et al.[5]presented a geometrical reasoning methodology for determining the optimal clamping points and clamping sequence for arbitrarily shaped workpieces. Liao and Hu[6]presented a system for fixture configuration analysis based on a dynamic model which analyses the fixture–workpiece system subject to time-varying machining loads. The influence of clamping placement is also investigated. Li and Melkote[7]presented a fixture layout and clamping force optimal synthesis approach that accounts for workpiece dynamics during machining. A combined fixture layout and clamping force optimization 3procedure presented.They used the contact elasticity modeling method that accounts for the influence of workpiece rigid body dynamics during machining. Amaral et al. [8] used ANSYS to verify fixture design integrity. They employed 3-2-1 method. The optimization analysis is performed in ANSYS. Tan et al. [9] described the modeling, analysis and verification of optimal fixturing configurations by the methods of force closure, optimization and finite element modeling. Most of the above studies use linear or nonlinear programming methods which often do not give global optimum solution. All of the fixture layout optimization procedures start with an initial feasible layout. Solutions from these methods are depending on the initial fixture layout. They do not consider the fixture layout optimization on overall workpiece deformation. The GAs has been proven to be useful technique in solving optimization problems in engineering [10–12]. Fixture design has a large solution space and requires a search tool to find the best design. Few researchers have used the GAs for fixture design and fixture layout problems. Kumar et al. [13] have applied both GAs and neural networks for designing a fixture. Marcelin [14] has used GAs to the optimization of support positions. Vallapuzha et al. [15] presented GA based optimization method that uses spatial coordinates to represent the locations of fixture elements. Fixture layout optimization procedure was implemented using MATLAB and the genetic algorithm toolbox. HYPERMESH and MSC/NASTRAN were used for FE model. Vallapuzha et al. [16] presented results of an extensive investigation into the relative effectiveness of various optimization methods. They showed that continuous GA yielded the best quality solutions. Li and Shiu [17] determined the optimal fixture configuration design for sheet metal assembly using GA. MSC/NASTRAN has been used for fitness evaluation. Liao [18] presented a method to automatically select the optimal numbers of locators and clamps as well as their optimal positions in sheet metal assembly fixtures. Krishnakumar and Melkote [19] developed a fixture layout optimization technique that uses the GA to find the fixture layout that minimizes the deformation of the machined surface due to clamping and machining forces over the entire tool path. Locator and clamp positions are specified by node numbers. A built-in finite element solver was developed. Some of the studies do not consider the optimization of the layout for entire tool path and chip removal is not taken into account. Some of the studies used node numbers as design parameters. In this study, a GA tool has been developed to find the optimal locator and clamp positions in 2D workpiece. Distances from the reference edges as design parameters are used rather than FEA node numbers. Fitness values of real encoded GA chromosomes are obtained from the results of FEA. ANSYS has been used for FEA calculations. A chromosome library approach is 4used in order to decrease the solution time. Developed GA tool is tested on two test problems. Two case studies are given to illustrate the developed approach. Main contributions of this paper can be summarized as follows:(1) developed a GA code integrated with a commercial finite element solver;(2) GA uses chromosome library in order to decrease the computation time;(3) real design parameters are used rather than FEA node numbers;(4) chip removal is taken into account while tool forces moving on the workpiece.3. Genetic algorithm conceptsGenetic algorithms were first developed by John Holland. Goldberg [10] published a book explaining the theory and application examples of genetic algorithm in details. A genetic algorithm is a random search technique that mimics some mechanisms of natural evolution. The algorithm works on a population of designs. The population evolves from generation to generation, gradually improving its adaptation to the environment through natural selection; fitter individuals have better chances of transmitting their characteristics to later generations.In the algorithm, the selection of the natural environment is replaced by artificial selection based on a computed fitness for each design. The term fitness is used to designate the chromosome’s chances of survival and it is essentially the objective function of the optimization problem. The chromosomes that define characteristics of biological beings are replaced by strings of numerical values representing the design variables.GA is recognized to be different than traditional gradient based optimization techniques in the following four major ways [10]:1. GAs work with a coding of the design variables and parameters in the problem, rather than with the actual parameters themselves.2. GAs makes use of population-type search. Many different design points are evaluated during each iteration instead of sequentially moving from one point to the next.3. GAs needs only a fitness or objective function value. No derivatives or gradients are necessary.4. GAs use probabilistic transition rules to find new design points for exploration rather than using deterministic rules based on gradient information to find these new points.4. Approach4.1. Fixture positioning principlesIn machining process, fixtures are used to keep workpieces in a desirable position for operations. The most important criteria for fixturing are workpiece position accuracy and workpiece deformation. A good fixture design minimizes workpiece geometric and machining accuracy errors. Another fixturing requirement is that the fixture must limit deformation of the 5workpiece. It is important to consider the cutting forces as well 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 of some of these problems [22].Common locating method for prismatic parts is 3-2-1 method. This method provides the maximum rigidity with the minimum number of fixture elements. A workpiece in 3D may be positively located by means of six points positioned so that they restrict nine degrees of freedom of the workpiece. The other three degrees of freedom are removed by clamp elements. An example layout for 2D workpiece based 3-2-1 locating principle is shown in Fig. 4.Fig. 4. 3-2-1 locating layout for 2D prismatic workpieceThe number of locating faces must not exceed two so as to avoid a redundant location. Based on the 3-2-1 fixturing principle there are two locating planes for accurate location containing two and one locators. Therefore, there are maximum of two side clampings against each locating plane. Clamping forces are always directed towards the locators in order to force the workpiece to contact all locators. The clamping point should be positioned opposite the positioning points to prevent the 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 are positive for all the time. Any negative reaction force indicates that the workpiece is free from fixture elements. In other words, loss of contact or the separation between the workpiece and fixture element might happen when the reaction force is negative. Positive reaction forces at the locators ensure that the workpiece maintains contact with all the locators from the beginning of the cut to the end. The clamping forces should be just sufficient to constrain and locate the workpiece without causing distortion or damage to the workpiece. Clamping force optimization is not considered in this paper. 4.2. Genetic algorithm based fixture layout optimization approachIn real design problems, the number of design parameters can be very large and their 6influence on the objective function can be very complicated. The objective function must be smooth and a procedure is needed to compute gradients. Genetic algorithms strongly differ in conception from other search methods, including traditional optimization methods and other stochastic methods [23]. By applying GAs to fixture layout optimization, an optimal or group of sub-optimal solutions can be obtained.In this study, optimum locator and clamp positions are determined using genetic algorithms. They are ideally suited for the fixture layout optimization problem since no direct analytical relationship exists between the machining error and the fixture layout. Since the GA deals with only the design variables and objective function value for a particular fixture layout, no gradient or auxiliary information is needed [19].The flowchart of the proposed approach is given in Fig. 5.Fixture layout optimization is implemented using developed software written in Delphi language named GenFix. Displacement values are calculated in ANSYS software [24]. The execution of ANSYS in GenFix is simply done by WinExec function in Delphi. The interaction between GenFix and ANSYS is implemented in four steps:(1) Locator and clamp positions are extracted from binary string as real parameters.(2) These parameters and ANSYS input batch file (modeling, solution and post processing commands) are sent to ANSYS using WinExec function.(3) Displacement values are written to a text file after solution.(4) GenFix reads this file and computes fitness value for current locator and clamp positions.In order to reduce the computation time, chromosomes and fitness values are stored in a library for further evaluation. GenFix first checks if current chromosome’s fitness value has been calculated before. If not, locator positions are sent to ANSYS, otherwise fitness values are taken from the library. During generating of the initial population, every chromosome is checked whether it is feasible or not. If the constraint is violated, it is eliminated and new chromosome is created. This process creates entirely feasible initial population. This ensures that workpiece is stable under the action of clamping and cutting forces for every chromosome in the initial population. The written GA program was validated using two test cases. The first test case uses Himmelblau function [21]. In the second test case, the GA program was used to optimise the support positions of a beam under uniform loading.5. Fixture layout optimization case studiesThe fixture layout optimization problem is defined as: finding the positions of the locators and clamps, so that workpiece deformation at specific region is minimized. Note that number of locators and clamps are not design parameter, since they are known and fixed for the 3-2-1 7locating scheme. Hence, the design parameters are selected as locator and clamp positions. Friction is not considered in this paper. Two case studies are given to illustrate the proposed approach.6. ConclusionIn this paper, an evolutionary optimization technique of fixture layout optimization is presented. ANSYS has been used for FE calculation of fitness values. It is seen that the combined genetic algorithm and FE method approach seems to be a powerful approach for present type problems. GA approach is particularly suited for problems where there does not exist a well-defined mathematical relationship between the objective function and the design variables. The results prove the success of the application of GAs for the fixture layout optimization problems.In this study, the major obstacle for GA application in fixture layout optimization is the high computation cost. Re-meshing of the workpiece is required for every chromosome in the population. But, usages of chromosome library, the number of FE evaluations are decreased from 6000 to 415. This results in a tremendous gain in computational efficiency. The other way to decrease the solution time is to use distributed computation in a local area network.The results of this approach show that the fixture layout optimization problems are multi-modal problems. Optimized designs do not have any apparent similarities although they provide very similar performances. It is shown that fixture layout problems are multi-modal therefore heuristic rules for fixture design should be used in GA to select best design among others.8Fig. 5. The flowchart of the proposed methodology and ANSYS interface.下載地址：https://doi.org/10.1016/j.compind.2005.05.0019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。102.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