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湖南工业大学 外文翻译专 业 机械设计制造及其自动化 学 生 姓 名 王 晓 雄 班 级 机本0303班 学 号 26030336 指 导 教 师 黄 开 友 MULTI-OBJECTIVE OPTIMAL FIXTURE LAYOUTDESIGN IN A DISCRETE DOMAINDiana Pelinescu and Michael Yu WangDepartment of Mechanical EngineeringUniversity of MarylandCollege Park, MD 20742 USAE-mail: AbstractThis paper addresses a major issue in fixture layout design:to evaluate the acceptable fixture designs based on several quality criteria and to select an optimal fixture appropriate with practical demands. The performance objectives considered are related to the fundamental requirements of kinematic localization and total fixturing (form-closure) and are defined as the workpiece localization accuracy and the norm and distribution of the locator contact forces. An efficient interchange algorithm is uaed in a multiple-criteria optimization process for different practical cases, leading to proper trade-off strategies for performing fixture synthesis.I. INTRODUCTIONProper fixture design is crucial to product quality in terms of precision and accuracy in part fabrication and assembly. Fixturing systems, usually consisting of clamps and locators, must be capable to assure certain quality performances, besides of positioning and holding the workpiece throughout all the machining operations. Although there are a few design guidelines such as 3-2-1 rule, automated systems for designing fixtures based on CAD models have been slow to evolve. This article describes a research approach to automated design of a class of fixtures for 3D workpieces. The parts considered to be fixtured present an arbitrary complex geometry, and the designed fixtures are limited to the minimum number of elements required, i.e. six locators and a clamp. Furthermore, the fixels are modeled as non-frictional point contacts and are restricted to be applied within a given collection of discrete candidate locations. In general, the set of fixture locations available is assumed to be a potentially very large collection; for example, the locations might be generated by discretizingthe exterior surfaces of the workpiece. The goal of the fixture design is to determine first, from the proposed discrete domain, the feasible fixture configurations that satisfy the form-closure constraint. Secondly, the sets of acceptable fixture designs are evaluated on several criteria and optimal fixtures are selected. The performance measures considered in this work are the localization accuracy, and the norm and distribution of the locator contact forces. These objectives cover the most critical error sources encountered in a fixture design, the position errors and the unwanted stress in the part-fixture elements due to an overloaded or unbalanced force system.The optimal fixture design approach is based on a concept of optimum experiment design. The algorithm developed evaluates efficiently the admissible designs exploiting the recursive properties in localization and force analysis. The algorithm produces the optimal fixture design that meets a set of multiple performance requirements.II. RELATED WORKLiterature on general fixturing techniques is substantial, e.g., 1. The essential requirement of fixturing is the century-old concept of form closure 2, which has beenextensively studied in the field of robotics in recent years 3, 4. There are several formal methods for analyzing performance of a given fixture based on the popular screw theory, dealing with issues such as kinematic closure 5, contact types and friction effects 6. A different analysis approach based on the geometric perturbation technique was reported in 7. An automatic modular fixture design procedure based on this method was developed in 8 to include geometric access constraints in addition to kinematic closure. The problem of designing modular fixtures gained more attention lately 9. There has also been extensive research in fixture designs, focusing on workpiece and fixture structuralrigidity 6, tool accessibility and path clearance 7. The problem of fixture synthesis has been largely studied for the case of a fixed number of fixture elements (or fixels) 8, 10, particularly in the application to robotic manipulation and grasping for its obvious easons 3, 4. This article aims to be an extension of the results on the fixture design issues previously reported in 14.III. FIXTURE MODELThe fundamental performance of a fixture is characterized by the kinematic constraints imposed on the workpiece being held by the fixture. The kinematic conditions are well understood 3, 4, 5, 7, 12. For a fixture of n locators (i = 1, 2, , n), the fixture can be represented by: dy=GTdqwhere define small perturbations in the locator positions and the location of the workpiece respectively. The fixture designis defined by the locator matrixi where and ni and ri denote the surface normal and position at the ith contact point on the workpiece surface. The problem of fixture design requires the synthesis of a fixturing scheme to meet a given set of performance requirements.IV. QUALITY PERFORMANCE CRITERIA FOR A FIXTUREA. Accurate LocalizationAn essential aspect of fixture quality is to position with precision the workpiece into the fixturing system. In general the workpiece positional errors are due to the geometric variability of the part and the locators set-up errors. This paper will focus only on the workpiece positional errors due to the locator positioning errors. As an extension of the fixture model equation (eq.1), the locator positioning errors dy can be related with the workpiece localization error dq as follows:Clearly, for given source errors the workpiece positional accuracy depends only on the locator locations being independent from the clamping system, the Fisher information matrix M = GGT characterizing completely the system errors. It has been shown 12 that a suitable criterion to achieve high localization accuracy is to maximize the determinant of the information matrix (Doptimality), i.e., max(det M).B. Minimal Locator Contact ForcesAnother objective in planning a fixture layout might be to minimize all support forces at the locator contact regions throughout all the operations with complete kinematic restraint or force-closure. Locator contact forces in response to the clamping action are given as: Normalizing these forces with respect to the clamping intensity we obtain:The force-closure condition requires these forces to be always positive for each locator i of a set of n locators:Computing the norm of the locator contact forces:leads to an appropriate design objective, i.e. minNote that this objective indicates both locator and clamp positions to be determined in the optimization process.C. Balanced Locator Contact ForcesAnother significant issue in designing a fixture is that the total force acting on the workpiece have to be distributed as uniformly as possible among the locator contactregions. If p represents the mean reactive force in response to the clamp action, then we define the dispersion of the locator contact forces as:Therefore, minimizing the defined dispersion represents an objective for a balanced force-closure: min(d).V. OPTIMAL FIXTURE DESIGN WITH INTERCHANGE ALGORITHMSAs mentioned earlier, by generating on the exterior surface of the workpiece to be fixtured a set of discrete locations defined as position and orientation, we create a potential collection for the fixture elements. For example, using the information contained in the part CAD model, a discrete vector collection (unitary, normal vectors) can be generated as uniformly as possible on those surfaces accessible to the fixture components (fig.1).Figure 1: Part CAD model and global collection of candidate locations for the fixture elements.The fixture design layout will select from this collection optimal candidates for locators and clamps with respect to the performance objectives and to the kinematic closure condition. Dealing with a large number of candidate locations the task of selecting an appropriate set of fixels is very complex.As already introduced in 12, 14 an effective method for finding the desired fixture with regard to one of the previous quality objectives is the optimal pursuit method with an interchange algorithm. Due to its own limitations and to the fact that the objectives are functions with many extremes, the exchange procedure may not end up to a unique optimized fixture configuration, but to several improved designs depending on the initial layout. Therefore the solution offered by the multiple interchange with random initialization algorithm is overwhelming favorable, fact that recommends this procedure over the single interchange algorithms. The algorithm can be described as a sequence of three phases:Phase 1: Random generation of initial sets of locators.The starting layout is generated by a random selection of distinct sets, each consisting from 6 locators out of the list of N candidate locations. If the clamp is pre-determined, avalid selection is obtained through a simultaneous check for all kinematic constraints. A big initial set of proposed ocators is preferred, giving the opportunity of finding a convergent optimal solution. However from the efficiency point of view the designer has to balance the algorithm between the accuracy of the final solution and the computation time.Phase 2: Improvement by interchange.The interchange algorithms goal is to pursue for an improvement of the initial sets of locators with respect to one of the objectives. Basically, this is done iteratively by exchanging one by one the proposed locators with candidate locations from the global collection. It is also essential to consider the form-closure restraint during the exchange procedure. The process will continue as long as an improvement of the objective function is registered. Studying the effect of interchange on the proposed quality measures leads us to some efficient algebraic properties. For example, an interchange between a current locator j (j = 1,2,6) and a candidate location k (k = 1,2, ,N-6) yields changes in the optimized function such that:Thus, at each interchange the pair is selected such that the significant term that controls the function evolution is improving, e.g. max p 2jk and min pc , easing the iterative process.Phase 3: Selecting the optimal solution.Applying the interchange algorithm for each initial set of locators we will end up with several distinct solutions on the configuration scheme of the fixture, the best fixture design corresponds evidently to the maximum improvement of the objective function. It should be emphasized that this algorithm can be used sequentially for different objective functions. Depending on the objective pursued the best solution can be evident (for a single objective) or might need the designers final decision (for multiple objectives).VI. MULTI-OBJECTIVE FIXTURE LOCATOR OPTIMIZATIONIn many applications the clamp is already fixed given some practical considerations. Then with the clamp predefined, the best fixture with respect to a certain performance criterion is constructed by selecting a suitable set of locators such that a significant improvement of the objective-function is registered. Using the random interchange algorithm we can analyze the impact of the optimization process on the fixture characteristics, as well as we can select the best optimized fixture solution for a specific criterion. In analyzing the effect of random interchange algorithm on several parts, there can be made the following statistical and empirical observations.A. Multi-objective trade-offsIn some applications both localization quality and a minimum force dispersion are important. In this case we may have to use a 2-step algorithm: first max(det M) and secondly min(d). The proposed order is a consequence of the above observations. First, maximizing the determinant will automatically decrease the dispersion. Next, a decreasing in dispersion leads in a decreasing in determinant value. Therefore, during the second phase of the algorithm tradeoffs between the two objectives occur. To solve the multi-objective optimization problem the interchange algorithm is applied successively for both objectives. With the clamp pre-defined, a rigorous check for form-closure is needed after each exchange step.A following set of plots present the results when the design requirements of precision localization and uniform contact forces are considered simultaneously. Fig. 2 and Fig. 3 illustrate the global changes of the fixture characteristics during the 2-step algorithm performed on an initial collection of distinct random sets of locators, with the clamp pre-fixed. It can be noticed the advantages of using max(det M) objective as a first step: while the determinant is increasing, the norm and the dispersion of the forces are decreasing, fact benefic for the overall quality of the fixture. Furthermore the solutions are convergent, such that the candidate set of locators for the next step will be significantly reduced. On the other hand, in the second phase, when applying min(d) optimization on sets of locators with a high determinant value the only trend in the determinant evolution is a decreasing one. Therefore, during the second phase of the algorithm tradeoffs between the two objectives occur, fact expressed also through the Pareto-line plot (Fig. 3). In this case the final decision has to be left for the designer to determine the best fixture scheme.Figure 2: Changes upon the fixture characteristics applying the 2-step optimization algorithm on an initial collection of random sets of locators.Figure 3: Behavior during a 2-step random interchange algorithm for a collection of locator sets.As an example, the behavior of a single initial set of locators is studied during the interchange processes of the 2-step algorithm (Fig. 4), confirming the previous remarks. The trade-off zone is decisive in the multiobjective design. The resultant configurations of the fixture after each successive phase are presented in Fig. 5. It can be noticed that the first objective moves the locators close to the boundaries as far as possible from each other, while the second one reorients them to the surfaces interior.Figure 4: General behavior of a 2-step interchange.Figure 5: Fixture configurations during a 2-step algorithm: (a) initial, (b) after max(det M), and (c) after min(d) respectively.B. Designer decision in finalizing the fixtureDuring the second phase of the algorithm a fairly significant decrease in the determinant value is registered, so few solutions will be acceptable for the multi-objective problem. In order to overcome these problems, an active designer control during min(d) interchange procedure is recommended. Essentially, the modifications consist in controlling the exchange procedure, such that the determinant of the improved locators must be permanently greater than a certain bound, simultaneously with the check for the form-closure condition. Even considering a tight bound for the determinant, more solutions are acceptable for the design than in the uncontrolled min(d) optimization case (fig. 6). As an example, the behavior of a single set of locators is studied during the interchange process of a 2- step algorithm controlled for two different bounds of the determinant value, emphasizing the fact that in the trade off zone the designer decision is decisive in finalizing the fixture configuration (fig. 7).Figure 6: Second phase of a 2-step random interchange algorithm: uncontrolled min(d); controlled min(d).Figure 7: General behavior during a 2-step algorithm applied on a single set of locators. (a) for B1 and (b) for B2.VII. OPTIMAL FIXTURE CLAMPINGThis section deals with a more complicated problem: to search simultaneously for the optimal clamp and locators in order to achieve a required fixture quality. Varying theclamp, it is obvious that the number of combinations for possible clamp-locators candidates is increasing very much. It will be shown that this problem is manageablefor the precise localization objective. For the other objectives we will have to restrain the search of the optimal clamp inside of a small set of proposed locations, such that the optimization procedure could be handled.A. Optimal Clamp from a Set of ClampsIn some applications the clamps have certain preferred locations, therefore the need to choose the best clamp from a proposed collection might be raised. For example, lets consider that a collection of preferred clamps is given, and an optimal fixture design with respect to the highly precise localization objective is needed. It is obvious that applying a random interchange procedure successively for each clamp, we find optimal fixture configurations for each specified clamp. Comparing the determinant values offered by these fixture schemes (fig. 8), we end up by selecting an optimal clamp and its corresponding locators, constructing the best- improved fixture design (fig. 9).Figure 8: Clamp selection from a collection of clamps for single-objective design.Figure 9: The initial collection of proposed clamps; the best clamp and the corresponding locators.B. Optimal Clamp from a Set of ClampsFurthermore, by extension, the selection of the optimal clamp from a set of proposed locations with regard to the multi-objective design problem can be considered. It consists of mainly applying the random 2-step interchange algorithm consecutively for each proposed clamp.By collecting the results after applying this procedure for all the clamps, we can compare their different behavior, and select the most appropriate one. It is obvious that an optimal clamp allows only small fluctuations of the determinant while the force dispersion is decreasing significantly (fig. 10). As an example, Fig. 11 illustrates the final fixture design consisting of the best clamp selected from a proposed collection with respect to the multi-objectives and the corresponding optimal locators.Figure 11: The initial collection of proposed clamps; the best clamp and the corresponding locators.VIII. CONCLUSIONSThis article focuses on optimal design of fixture layout for 3D workpieces with an optimal random interchange algorithm. The quality objectives considered include accurate workpiece localization, minimal and balanced contact forces. The paper focuses on multi-criteria optimal design with a hierarchical approach and a combined-objective approach. The optimization processes make use of an efficient interchange algorithm. Examples are used to illustrate empirical observations with respect to the design approaches and their effectiveness. The work described here is yet complete. Since the inter-relationship between the locators and the clamps has a determinant role on the fixture quality measures, a more coherent and complete approach to study the influence of the clamp and search of the optimal clamp position is needed in future works.IX. REFERENCES1 P. D. Campbell, Basic Fixture Design. New York: Industrial Press, 1994. 2 F. Reuleaux, The Kinematics of Machinery. Dover Publications, 1963.3 B. Mishra, J. T. Schwartz, and M. Sharir, On the existence and synthesis of multifinger positive grips, Robotics Report 89, Courant Institute of Mathematical Sciences, New York University, 1986.4 X. Markenscoff, L. Ni, and C. H. Papadimitriou, The geometry of grasping, International Journal of Robotics Research, vol. 9, no. 1, pp. 61-74, 1990.5 Y.-C. Chou, V. Chandru, and M. M. 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Yuan, A variational method of robust fixture configuration design for 3-d workpieces, Journal of Manufacturing Science and Engineering, vol. 119, pp. 593-602, November 1997.11 D. Baraff, R. Mattikalli, and P. Khosla, Minimal fixturing of frictionless assemblies, CMU-RI TR-94-08, The Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, 1994.12 M. Y. Wang, An optimal design approach to 3D fixture synthesis in a point set domain, IEEE Trans. on Robotics and Automation, December 2000.13 A. Atkinson and A. Doney, Optimum Experimental Designs. New York: Oxford University Press, 1992.14 M. Y. Wang and D. Pelinescu, Precision localization and robust force closure in fixture layout design for 3D workpieces, IEEE Intl Conf. on Robotics and Automation (CD-ROM), San Francisco, April 2000.在独自领域最佳多功能夹具布置的设计 Diana Pelinescu and Michael Yu Wang 马里兰大学机械工程系 College Park, MD 20742 USA摘要：本文论及一个在夹具布置设计的重要问题: 根据实用要求来选择一套优化的装置，并评估可接受的装置设计。主要宗旨是要考虑与根本要求有关的运动学方面和总的夹具形式(形式闭)， 还要考虑作为工件定位的准确性和标准以及定位布置的联系。 高效率的互换算法被使用在多标准优化过程有很多不同的实用案件, 因此合适的交换平台能执行综合装置。关键词：夹具 定位器 互算法 目标函数1介绍适当的夹具设计对产品质量方面精确度和准确性在部份制造和装配是关键的. 夹具 系统, 通常包括钳位和定位器, 必须是可胜任保证成品质量 包括安置和保证在机器生产中的加工过程.虽然有几个设计指南如3-2-1规则, 自动化的系统为设计夹具提供的CAD 模型在演变。这是一篇描述自动设计夹具之类的3D研究方法文章. 这部分被认为是固定当前任意复杂几何, 并且被设计的夹具必需限制在对元素的最小数字, 即六台定位器和钳位。此外,固定器被限制在无非摩擦点接触并且实用的位置. 总之, 夹具之类必需放在假定的潜在准确的位置; 例如, 位置也许由可分辨的工件外表面决定的。首先确定夹具设计的目的, 从被提出的的特殊领域,在这个领域可行的夹具设计必需满足形式关闭限制。第二,可接受的夹具设计类被评估在几个标准然后选择最理想的夹具设计。在这工作中是地方化准确性、和定位器是被考虑成工作指标的准则和效率。这些标准包括在一个夹具设计中的关键错误来源、位置误差和不需要的压力在部份夹具原理，这种原理是由一个被超载的或被失衡的力量系统决定的。优选的夹具的设计方法是建立在最宜的实验设计的概念的基础上的。被开发算法高效率地利用递归物产在定位和力量分析评估可接受的设计。算法产生符合一套多个性能要求的优选的夹具设计。2 相关工作本文大概的夹具设计技术是可替代的,如1 。夹具设计的根本要求是形式关闭 2 的旧概念, 近年来广泛地被运用在机器人学 3, 4 研究在领域的。有几个正式方法根据普遍的螺丝理论为分析一套指定的夹具的性能, 应付问题譬如连接类型和摩擦作用 6 的运动学关闭 5 , 另外一种根据几何学扰动技术分析方法被报告了 7 。根据这个方法自动夹具模具设计程序被开发 8 包括限制除运动学关闭之外的几何学通入。夹具模具设计的问题最近获取了更多注意 9 。现在对夹具设计也有了广泛的研究, 集中于工件和夹具结构硬度 6 , 工具可及性和轨道清除 7 。夹具设计的问题主要是研究夹具元素(或夹具定位) 8, 10 一个固定的数字的事例, 特别在应用对机器人操作和掌握是它明显的原因 3, 4 。这篇文章关于引伸夹具设计问题的目的是早先被报告 14 。3. 夹具模型 夹具的根本表现为限制工件运动既描绘强加给工件由夹具限制。运动学条件很好被了解 3, 4, 5, 7, 12 。一个夹具有n个定位(i = 1, 2., n), 夹具可能表示： TMy=GTTMq(1) TMy TMy TMyTMyn12=TMq=TMr TM T定义了小扰动在定位位置和定义工件位置。夹具设计由定位器矩阵G=h1h2hn=即 hiT=-niT(ri*ni)T定义并且ni 和ri 表示表面法线和安置在ith 接触点在工件表面。夹具设计的问题要求一份夹具的计划的综合指定的一套性能要求。4. 夹具的质量表现标准A定位准确化夹具质量的一个根本方面是安置工件入夹具系统的精确度。总之工件位置错误归结于零件和定位器设定错误的几何学可变性。本文将集中于工件位置错误是由于定位器引起的位置误差。作为工件式样等式的引伸(例1), 定位器位置误差TMy可能工件定位错误TMq 如下： TMy =TMqT(GGT)TMq 2 (2)明显的, 为指定的来源错误的工件位置定位准确取决于定位器位置是一个独立与夹紧的系统,夹板信息矩阵M = GGT完全地描绘了系统的误差。它显示了 12 是一个标准的标准达到高定位化准确性将最大化信息矩阵, 即, max(det M). 的定列式。B.触点压力的最小定位其他夹具布局定位的设计也许将使所有在定位器触点位置支持力减到最小，在所有操作过程中完全以运动学克制力锁合。定位触点力以回应夹紧的行为给定：我们获得这些力正常夹紧的强度:力锁合情况要求这些力量是总是正面的为各台定位i 和一套n 定位器:计算准确触点力定位准则:导致一个适当的设计标准, 即分钟 。注意这个标准表明了定位器和钳位位置被确定在优化过程。C. 平衡的触点力定位。其它夹具设计中的重大问题是, 力总和在工件表现为成尽可能均匀地被分布可在定位器触点位置之中。如果代表易反应的力以回应钳位运动, 我们然后定义定位器触点力的分散作用如:所以, 减小最小离散表示为平衡的力锁合定义一个标准为: 。5.用互算法优选夹具设计依照前面提到, 在工件的外部表面设计夹具是一套独自定位作为位置和取向,我们定义了一套潜在的夹具元素。例如, 使用一些信息包含零件CAD 模型, 独自的矢量汇集(单一正常的矢量)， 尽可能均匀地引起在那些表面容易接近对夹具压力(fig.1) 。图1: 为夹具元素分开CAD 模型和球形集。夹具设计布局从这优选集挑选成为为定位器并且钳位，这些涉及表现标准和对运动学关闭的适应。为很大数量的选择物定位选择适当固定方法的是非常复杂的。依照在 12, 14 已经被介绍的一个有效方法为找一个想得到的夹具关系到早先质量标准，以互换算法选择当中一个是优选的选择方法。由于它自己的局限和对事实标准的作用有许多弊端, 交换做法不能结束由一种独特的优化决定夹具配置, 但根据最初的布局改善设计。所以由多互换提供以任意初始化算法的解决方案非常有利的， 推荐互换算法的这个做法在唯一途径。阶段1: 最初的随意可替代的定位器。开始的布局是集合的一种任意选择, 每个引起包括从6 台定位器在N选择物定位。如果钳位被预先决定, 获得合理的选择的方法是通过一次同时检查对于所有运动学限制。跟趋向于选择一套最初提出的定位, 它提供了发现一个会聚最佳方案机会。但是从效率观点来看设计师必须在最后的解答的在算法准确性和计算时间之间寻求平衡。阶段2: 由互换来改善平衡。互换算法的目标是追求为一套方案选择一个定位达到最初的标准。基本上, 这由逐个交换重复做从而从球集找出的定位器为选择物定位。它也是在交换程序中根本考虑形式关闭克制。过程将继续改善只到目标函数定位。建立在提出的质量的措施研究互换的作用将带给我们一些高效率的代数产物。例如, 互换在一台当前的定位器j (j = 1,2.,6) 和候选定位器k = 1,2.,N- 6) 之间在优化作用上生这样变化:因而,各互换对被选择, 这样控制演变得到重大改变，即最大pjk2，缓和迭代过程。阶段3: 选择最佳方案。 为每个最初的定位运用互换算法我们将最终获得确切的解决夹具结构配置方案，最佳的夹具设计的配置计划显然地反映目标函数的最大改善。 需要强调的是这种算法可能连续地被使用为不同的目标函数。根据设计目标选择最佳的解答是显然的(有一个唯一目标函数) 或许需要设计师的最后的决定(有多个目标函数) 。6.多目标夹具定位器优化在许多应用上夹板已经是固定的被给予一些实用考虑过。然后如果钳位被预定义, 最佳的夹具涉及某一表现标准由选择适当的套定位器, 这样明显作用的重大改善被认可。使用任意互换算法我们分析夹具表面最佳的压力，这样我们能选择最佳的夹具设计方案标准。通过在几个零件中使用分析任意互换算法的作用,可以使用以下统计和经验的观察。A. 多目标平台在一些应用中定位质量和极小的力的分解作用都是很重要的。在这种情况下我们需要使用2 步算法：第一最大（det M）第二最小（d）。提出的命令是上述观察的结果。首先, 最大化定列式自动地将减少分散作用。其次, 减少在分散作用导致减少定列式值。所以, 算法的第二个阶段期间在二个目标函数之间发生。解决多目标优化问题互换算法需要连续这两个目标。一旦钳位被预定义, 一次严谨检查对于形式关闭是需要的在各交换步骤以后。当需要定位精确和压力一致时由一系列的结构可以得出结果，设计要求同时要被考虑。图2 和图3 说明夹具特征在2 步算法期间球性变动时执行的最初集在任意套的定位, 与钳位被提前定位。首先必须注意到使用最大(det M) 目标函数的优点。当定列式增加时, 准则和力量的分散作用是越来越少的, 事实上对夹具的整体质量是有利的。而且解决方案是集中的, 这样为下步预选的套定位器将显著减少。另一方面, 在第二个阶段,当在一系列定位器中应用min(d) 最优方案需要高定列式价值，唯一的趋向在定列式演变是逐一减少的。所以, 在算法平台的第二个阶段期间在二个目标函数之间发生, 事实并且被表达通过Pareto 线剧情(图3) 。在这种情况下最后的选择必须希望设计师确定最佳的夹具方案。例如,研究一套初始定位器的特定性能可以证实以前在二步算法过程中的互换过程。平衡区域是在多目标设计中起决定性作用。夹具的合力分配在连续部分如图5 。它可能被注意, 第一宗旨从彼此移动定位器紧挨界限尽可能的, 当