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自动装配模型注塑模具X. G. Ye, J. Y. H. Fuh 和K. S. Lee新加坡国立大学机械和生产工程系注射模具是包含产品生产部分和自动化装配的组件，本篇论文论述了注射成型模塑的两个关键的装配组件，也就是由计算机设计模拟出来，并且决定非生产部件在装配组件中的位置和取向。这种从本质特征和客观取向所设计的组件，是为了表现出注射成型装配组件的。这种设计允许设计者忽略模塑制件的细节，直接描述制件重点的那一部分及其原因，因此它给设计者提供了一次设计装配的机会，一个系统的简单几何方法通常可以在相同的条件下推断出一个客观配件在装配组件中的位置。在这种粗略设计和系统简单的几何方法的基础上，模塑自动化装配组件被进一步研究。关键词：模塑组装，表观现象，注射成型，客观取向1. 前言 注射成型是塑料制造业中最重要的一个环节。它所必要的设备包含两个部分：注射成型机和注射模具。今天所用的注射模具机就是所谓的万能机，在一定的范围内不同尺寸的塑件通过它生产出来，但是模具设计时需要根据塑件的要求进行改变。不同的模具布局对于不同的模具形状是必要的。注射成型模具的最基本的任务就是将熔化了的材料生产成不同形状的制品，这个任务是由包含了阳模、型腔、嵌件、一级顶出机构的型腔系统完成的。型腔系统的几何形状和尺寸是由塑件直接决定的，因此一个型腔所有的构件称为产品生产部件（产品就是指塑件，部件就是指注射模具的构件），另外产品成型是最基本任务，注射成型机需要完成很多任务，例如，分配、熔料、冷却融化物注射塑件，功能部件所完成的这些任务与注射成型不同结构和尺寸的塑件非常相似，他们的结构和几何形状与塑件成型模具不相关，但是他们的尺寸可以根据塑件的尺寸而不断改变，因此我们可以从中得到结论：注射成型机包含了塑件生产部分和与之相关的非生产部分的自动化装配组件。图1所示为注塑模具的装配机构：图1 注塑模具的装配机构设计作为产品依赖的一部分是基于塑件的几何形状，最近几年，CAD/CAM技术已经被成功的应用在帮助模具设计者设计塑件生产部分。塑件的自动化生产作为塑件生产的一部分给人们带来很大的研究兴趣，然而在塑料模具装配建模上却很少有行动，尽管它和产品的设计一样很重要。当应用CAD系统设计产品生产部分和所有的注射成型装配组件时模具工业面临以下两种困难，第一，在一个模具系统中往往有一百多个产品生产部分，并且这些部分之间互相联系互相制约，模具设计者在一个装配组件取向和安排这些组件时浪费了很多时间，第二，模具设计者利用大部分的时间来思考实际存在的客观部件的选用原则，例如，螺丝、底座、钉子时，CAD系统应用了一个完全不同的客观无体几何水平。结果高水平的客观取向想法不得不翻译成低水平的CAD系统，例如，线、面、块，因此，对于解决上面的两个问题，发展一个模塑自动化装配组件是非常重要的。在这篇论文中我们论述了模塑自动化组建的两个关键因素。在计算机中涉及模具生产部分和模具生产装配组件，并且决定组成部分在一个装配组件中的位置和客观取向。这篇论文简要的描述了模塑装配组件的相关研究，并且论述了注射成型模塑装配组件的一个不可缺少的设计，一个简单的几何方法被用来决定一个部件在模具装配组件中的位置和客观取向，介绍了注塑成型模塑的自动化装配组件的一个例子。2相关的研究模具装配已经在很多领域广泛研究，例如，动力机体学、人工智能和几何建模学。Libardi等人编辑了一本模塑组件的书，他们在其中报道了许多研究者在模具装配研究中利用图标机构。在这个图标方案中，一些组件被比喻成鼻子，一些感观机体被弧线连接起来。然而，这些感官机体并没有重合在一起，这就严重的影响了改性过程。例如，一个几何装配移动，所有与之相关的部件没有相应的移动。Lee 和 Gossard发明了一种支持包含了最基本信息的数据库的有等级差别的装配组建系统，这些最基本的信息包含了两个部件之间的垫片，这些改型基体取决于与之相关的实体，但是这些有等级模具差别的模具组建仅仅代表了这些模具中的一部分。自动化推断配置部件在装配中的作用，意味着模具设计者可以避免直接定义这些改性机体，另外，一个部件位置的改变，将会随着与之相连的任意一个部件的形状和位置的改动而变动，存在着三种技术推断计算一个部件在一个装配组件中的位置和取向，这三种技术分别是数字迭代技术、系统代数技术、和系统几何技术。 Lee和 Gossard指出数字迭代技术使用计算存在与空间关系中任何一个部件的位置和取向，他们的方法包含了三个步骤：产生约束等式、减少约束等式的数量和解决这些等式，存在着16个等式与条件不符，18个等式符合条件，6个对任何机体合适的等式，还有2个附加等式符合旋转部分，通常这些等式的数量超过了可以利用的等式，因此这就需要一种技术筛选到不需要的公式，牛顿力学公式被用来解决这个问题。这种技术有两个缺点：第一、这种方法严重的依赖这前面的方法；第二、数字迭代技术不能区别不同的数字基础，因此，它很可能用在空间关系的问题上，这个领域不是数学方面的空白，但是在理论上还很模糊。Ambler 和Popplestone提出了一种方法用来计算装配组建中的每一个部件在两个部件之间转换和改观方面所需要的空间关系，每一个部件存在着关于空间关系的六个转变三个移动和三个旋转)。这种技术需要大量的计算机程序应用和数据计算，同样他不能用来解决在任何时间出现的所用问题，尤其是当一个等式不能在程序中被重写的时候。Kramer发明了一种能够决定一个刚性物体的位置和取向的集合方法，这种方法满足一套的几何约束。这种几何方法是通过产生一系列没任何约束的系统方法来解决问题的。这就导致了DOF数量的下降，Kramer利用了一种解决问题的技术称作“上帝”，这是一个包含了一个点和两条直交轴线的技术。7条约束现在图标间被定义出来，对于一个涉及到单个对象与约束在那个机构上的标记与不变属性和实验分析的标记的问题获得了解决。经试验分析后确定一个几何物体的最终布局，再一步一步的解决客观物体的布局，自由度决定哪一种方法将会满足一个物体没有约束，这考虑到将来减小物体的自由运行，在每一步的结尾，一个可行的的法案增加到装配计划中。根据Shah和 Rogers的方法，Kramer的描述在塑装配组建技术中起到了重大的作用，这种系统的几何方法可以解决所有的约束条件，并且与数字迭代技术相比他拥有更吸引人的数字计算技术，但是要应用这种方法就需要大量的应用程序。尽管很多的设计者积极的投入到模塑装配组建技术上，但是关于塑料注射成型模塑装配组建技术的成果很少被系统的报道。Kruth等人发明一种支持注射成型的设计系统，他们的系统通过高水平的功能模塑组建支持了注射成型模塑技术，因为他们的技术是建立在AUTOCAD的基础上，所以只能通过简单的块和线框表示出来。3注射成型装配组件的表征注射成型模塑的自动化装配组件的两个关键技术是在计算机中将模塑装配组件表示出来以及决定部件的生产部分在装配组件中的位置和方向。在这个阶段我们可以利用客观取样和表观现象来代表注射成型的装配组件。在计算机中会使每个部件的结构与空间关系变为现实，这种设计必须支持所有的部件在装配组件中的配合,所有部件间的改变关系和装配组件作为一个整体的操作要求。另外，装配组件的这种设计要求设计者在设计师必须满足以下的要求：1、要求模具设计者有高水平的技术来应对实际存在的物体水平；2、装配组件的这种设计必须能够正确的表现出自动生产过程的功能；为了满足这些要求一种具有表观现象和客观取向的有等级差别的模具被应用在注射成型技术中，一个装配组件可以被分成很多集合装配，这个集合装配有包含了很多的构件，因此有等级差别的模具能够分成合适的代表两个构件之间的结构。一个等级差别的模具暗示了一个明确的装配组件组，另外，一个等级差别的膜具能够直接表现出一个构件对另一个构件的依赖。基于特征的设计要求设计者站在一个比实际应用模具更高的水平线。几何形状是直接的、有尺寸的，当功能建模设计出来细节，使用者能够很快的通过一系列参数直接定位。当然，由于几何形状之间的关系，它同样也可以使设计者比客观取向的模具设计者容易做出变动。没有客观取向设计者就要根据模具要求考虑到所有的几何结构，因此，每一次设计改变可直接根据模具要求改变而改变。另外，客观取向的设计能够给设计者提供更高水平的组件物体，例如，当模具设计者考虑到物体的真实水平时，如一个冷却水孔相关的客观表象在计算机中模拟使用。客观取向模具设计是一个在考虑到现实事件中的模具技术的基础上所应用的一种新的方法，它的基础是客观物体，这个物体结合了数据结构和性能客观取向方法被用来理解问题和设计程序和数据库，另外，客观取向代表了装配组件制造时单位客观物体和总的组件的包含与被包含关系。图2表示了模具装配中的表观现象与客观取向，代表性就是抽象的层次结构来自低水平的几何实体到高水平的组件。项目图包含方格代表装配模具，实线代表部件的关系，虚线代表其它的关系，部件组成实体。它联合了表观现象与客观取向的优点，它不仅包含部件与整体的关系，更拥有一些模具装配的结构关系和使用功能。在3.1中将进一步分析模具装配的意义，而在3.2中详细的介绍各部件之间的关系。图2 模具装配中的表观现象与客观取向3.1装配模具的定义在我们的工作中，把模具装配“O” 在下面的公式中是独一无二的实体：O = (Oid, A, M, R) (1)在这里：Oid是模具装配的唯一标识符(O)；A是一个三元组合(t，a，v)， 其中a是O的一个属性，每个与之相关的元素就是字符t, a和v；M是一个数组，(m, tc1, tc2, %, tcn, tc)。每一个元素都是每个功能的唯一标识符。R是O与其它元素之间的关系，这儿又六种基本的关系在模具装配中。即：Part-of, SR, SC, DOF, Lts, and Fit.3.2装配模具的关系在模具的装配中有六种关系，分别是：Part-of, SR, SC, DOF, Lts, Fit。Part-of模具装配本身的一种性质；SR明确规定的方向与装配部件的客观取向之间的关系，对于一个组成部分，它们特殊的空间关系来源于特殊的限制(SC)；SC 一个部件与其它的装配组件之间的限制关系；DOF 在装配完成后允许的旋转自由度；Lts 由于旋转的自由度，单方向或多方向的限制；Fit 尺寸的限制为了保持某种状态；在模具装配的所有元素中，模具的装配各部件之间的关系是非常重要的，这种关系不仅决定部件的方位，而且能够保持模具装配的关系。接下来，我们将要进一步通过例子来阐述这种关系。3.2.1形状特征间的关系从本质上说，模具设计就是一种思想，模具设计师们大部分时间在想一些实际的东西，例如底板、螺丝钉、沟槽、斜面和孔。所以，非常的有必要利用所有的标准部件建立几何模型。模具设计师可以很容易的对部件的方位以及形状进行改变，因为那些部件间的关系都显示在模型上。图3a显示一个平板上有个孔，这个部件有两个特征：阶段性和反钻孔，反钻孔(FF1)在F2和 F1的坐标系中可用FF1表示，公式(2)(5)表示了二者之间的关系。对于形状特征，这里没有什么特殊的限制，所以设计者直接指定这种关系。这种详细的关系如下：图3 装配关系公式(2)-(7)显示了FF1与FF2之间的关系，这些关系最终决定在这一部件中的方位和方向，如果把这一部件看做是模具装配，形状特征便是模具装配中的一个“元件”。形状特征的选择是建立在标准零部件的尺寸的基础之上的，因为CAD/CAM系统技术的支持，形状特征不仅满足模具装配的尺寸和位置，而且体现了我们选择的模具部件之间的特殊关系。为了增加那种特殊的关系，我们必须记录逻辑关系，为模具装配配备合适的关系。在CAD系统更新前， 检查合适的形状特征都是必要的。 3.2.2各部件之间的关系 在模具装配中，某部件的方向和方位是与其它部件相关联的。图3b显示了一个模板(PP1)与螺丝钉(pp2)。螺钉的相对位置被模板上的孔所限制，这种模板与螺钉之间的关系如下：就像我们从公式(8)和(9)所看到的一样，通过计算Mp 和Mr决定螺钉在平板上的位置和方向是非常重要的。Mp 和Mr来自空间的限制，这种推导需要推断模具装配中的配置，这个问题将在下一阶段进行讨论。我们已经介绍了一种注塑模具装配的计算机模型。在这一阶段，总结计算机模型的长处是有必要的。装配模具作为组件的代表，反过来却包括了组件，这个组件可以进一步代表装配的形状特征，如此等级关系暗示装配顺序以及包含关系。这种基础特征不仅要求设计师们能够高水平的设计单个部件，而且扩展了模具装配的功能，因为这种性质允许部件随同其他部件位置和方向的改变而改变。表观现象可以联合某个对象的数据库以及操作。模具装配中封装的功能就像顶出机构与干涉检查一样能够使的例行程序自动化。4注射成型模塑的装配部件推断我们从等式8和9中可以看出，装配模具中部件的方位与方向最终会被变换母体所取代。为了方便，特别的机构关系通常是一些高水平的交配条件，就像伙伴、共线和平行。所以，在部件之间的隐含约束关系间自动获取明确转变条件是很必要的。在第二部分中已经对三种推断模具装配部件的技术进行了分析。因为几何模型能够找出所有问题，并同时间的复杂性组成等式，我们用这种方法去确定装配模具部件的方位与方向，要将这种方法在模具装配中付之于行动，需要大量的规划。然而，一种简化几何的方法被提议去确定模具装配时的部件方位与方向。在这种简化几何的方法中，通过产生一系列的行动去满足那些约束来确定模具装配时的部件方位与方向。满足约束的所需资料被存储在“计划碎片”中，每个计划碎片就是一个能够指定顺序和能够使移动部件按照预定的路线移动的程序，计划碎片还可以记录物体的新自由度和相关的几何变量。从概念上说，Kramer的计划是一个三维调度表。我们用TDOF来代表自由度的平移度，RDOF来代表自由度的旋转度。这种计划表详尽的例举了所有空间的状态，为了满足一系列的约束在物体上的标记和球坐标系中的标记而移动物体。例举了上面三个因素相结合的不同，将导致82中差异。如果研究的空间减少的话，将会降低计划表中的差异。为了实现这一目标，那些因素的例举数量需减少。例如，对于一定的约束类型，如果TDOF由0,1,2 ,3变为0,3后，这样研究空间就严重的降低。经过仔细的分析模具装配部件的组成，四个基础的约束被引用，他们的定义和代数方程如下：由于那些相关的限制，我们计划表中的参数充分的下降了，在我们的计划表中去解决一个、两个和三个的限制，必须有九个参数。模具装配中为了互相增加组件，更多的限制约束和自由度将会增加使用者的适应性。然而，在自动化模具装配中，许多特殊的模具装配关系将会事先确定，许多顺序关系也不太重要。通过上面的约束限制，各部件之间的结构关系将会被资料库中数据指定。当增加模具装配中增加部件时，系统将会第一时间分解复杂的约束为简单的约束，然后，生成一组碎片计划使部件在模具装配中有确定的方向和方位。5模具非标准的集合装配许多注塑成型的装配组件包含了生产部分和非生产部分, 生产部分单个组件的设计是在塑件几何性能的基础上得到的，通常产品生产部分应有与高水平装配组件相同的取向，通常这些构件的位置和大小直接由设计者制定，对于产品省成本的设计，传统的方法是设计者从手册中直接选择所需要的模型，对这些选择的产品生产部分建立几何模型块，然后再将这些块加入到注射成型装配组件中,这样的设计既浪费时间又错误百出。在我们所应用的新的设计方法中，所有生产部分的设计数据是在装配组件和客观需要的基础上得到的，这些数据不仅包含了几何形状和生产部分的尺寸，而且还包含了部件之间的空间约束。另外，许多部件有固定的路线，例如：顶出和复位同样也在这个数据库中，因此，模具设计必须选则应用这要求的模具生产部分而确定它的结构，然后计算机软件会自动的确定出这些部分所要求部件的取向和位置，最后再将这些部件加入到装配组件中。5.1模架组件就像我们从图1中看到的那样，产品的生产部分可以进一步分为标准件和非标准件。这些非标准件是由一系列底座、导向机构等装配组件构成的。除了确定产品的性状外，一个模具还必须同时完成许多其他的功能，例如冷却、注射产品、顶出、合模导向等，许多的模具应有相似的性能，因此这就导致了他们在结构上的相似，模具结构设计中有许多标准要求，模具非标准部件就是在这些标准组件的基础上设计而成的。根据装配组件设计师的客观取样和表观现象，模具组件的表观现象是非标准件首先应该考虑到的，另外，客观组件的设计受到组件构件之间的相互关系和构件的功能的限制，然后利用这些客观组件，一个有等级差比的集合装配(模具的非标准部件)就形成了，这些模具的生产部分可以直接由目录数据库中的数据确定，图4表示模具实际情况对立于模具的特殊组件，这种特殊的模具组件的例子被自动的添加的模具装配中，模具部件与整体之间的结构关系可以以Mp和Mr为单元表示出来。图4 模具设计实例5.2标准件的自动添加一个标准件就是一个自动组件，这个可以依据3.1中的公式1定义。在数据库中的空间约束是由交差、平面直线和弧线确定的，但是它与非标准件不同，标准件的位置和客观取向并不确定。在设计时软件通过简单的公式直接推断出标准件的几何形状。5.2模具装配中的复位机构自动化设计的一个关键考虑因素是复位过程，复位是指嵌件在设计过程中留出相应的一定空间使之归位的操作。当一个顶出设备加入到装配组件中就要求在设计过程中留出相应的孔，以便复位。如图5所示：图5 模具顶出机构既然利用了客观取向技术，每一个装配组件都可以有两种表示方式：实际存在和客观设计。实际存在的物体空间是根据一个真实物体所要占据的空间决定的，无论何时一个客观构件被加入到装配组件中，它的真实空间尺寸也同时被设计出来，复位操作技术是根据相关构件的相互关系而设计的。另外由于实际空间和真实空间的关系，复位技术的设计也要根据实际物体做出相应的改变，这种自动复位功能进一步说明了客观取向的优点。6系统实施在客观取样和表观取向基础上设计而出的模塑自动化装配组件技术，已经在美国国立大学被应用在IMOLD领域内。这种绘图技术是提高应用程序的一个有效方式，通过这种技术的使用这可以将其它部分加入到装配组件中修改参数等。尽管绘图技术提供了很好的功能，但是上文提到的方法仍然被用来推断构件的布局，因为在设计过程中必须考虑到构件自由运行的程度和检查构件在加入到装配组件以前所需要的空间。这种系统的约束条件和图表约束是相辅相成的。图6显示为一种注塑产品，图7a中显示成型这种塑件的注塑模具，图7b中指出了各个组件在装配时的父子关系。这种装配组件使用IMOLD技术设计的，模具中的每一个非标准件被自动的安置在装配组件中。同样，标准件(例如：螺钉)也是被自动的加入到装配组件中,复位技术也不例外。图6 注塑产品图7 成型注塑产品6的模具7. 结论在表观现象和客观取向技术上所设计的具有等级差别的注射模塑装配组件，不仅仅提高了装配组件设计技术，而且同时提高了操作功能和几何约束性，例如：自由的程度,配合条件、镶嵌和取向约束。因为，装配组件设计的这一技术的提高，装配组件中某一构件的尺寸变化可以在整体设计完以后再做出变动。装配组件构件的封装有以下两个特点：1、配合技术在装配构件中封装自动化装配组件设计时可以被容易的利用；2、装配组件的封装使得装配组件的设计在应用过程中自动完成，例如：复位和构件检查。提出的简单的统计方法可以直接降低自动化设计过程中程序的难度。鸣谢作者感谢新加坡国立大学和新加坡国家科学技术委员会对IMOLD研究项目的支持。第 12 页 共 12 页Int J Adv Manuf Technol (2000) 16:739747 2000 Springer-Verlag London LimitedAutomated Assembly Modelling for Plastic Injection MouldsX. G. Ye, J. Y. H. Fuh and K. S. LeeDepartment of Mechanical and Production Engineering, National University of Singapore, SingaporeAn injection mould is a mechanical assembly that consists ofproduct-dependent parts and product-independent parts. Thispaper addresses the two key issues of assembly modellingfor injection moulds, namely, representing an injection mouldassembly in a computer and determining the position andorientation of a product-independent part in an assembly. Afeature-based and object-oriented representation is proposedto represent the hierarchical assembly of injection moulds.This representation requires and permits a designer to thinkbeyond the mere shape of a part and state explicitly whatportions of a part are important and why. Thus, it providesan opportunity for designers to design for assembly (DFA). Asimplified symbolic geometric approach is also presented toinfer the configurations of assembly objects in an assemblyaccording to the mating conditions. Based on the proposedrepresentation and the simplified symbolic geometric approach,automatic assembly modelling is further discussed.Keywords: Assemblymodelling;Feature-based;Injectionmoulds; Object-oriented1.IntroductionInjection moulding is the most important process for manufac-turing plastic moulded products. The necessary equipment con-sists of two main elements, the injection moulding machineand the injection mould. The injection moulding machines usedtoday are so-called universal machines, onto which variousmoulds for plastic parts with different geometries can bemounted, within certain dimension limits, but the injectionmould design has to change with plastic products. For differentmoulding geometries, different mould configurations are usuallynecessary. The primary task of an injection mould is to shapethe molten material into the final shape of the plastic product.This task is fulfilled by the cavity system that consists of core,cavity, inserts, and slider/lifter heads. The geometrical shapesCorrespondence and offprint requests to: Dr Jerry Y. H. Fuh, Depart-ment of Mechanical and Production Engineering, National Universityof Singapore (NUS), 10 Kent Ridge Crescent, Singapore 119260.E-mail: mpefuhyhK.sgand sizes of a cavity system are determined directly by theplastic moulded product, so all components of a cavity systemare called product-dependent parts. (Hereinafter, product refersto a plastic moulded product, part refers to the component ofan injection mould.) Besides the primary task of shaping theproduct, an injection mould has also to fulfil a number oftasks such as the distribution of melt, cooling the moltenmaterial, ejection of the moulded product, transmitting motion,guiding, and aligning the mould halves. The functional partsto fulfil these tasks are usually similar in structure and geo-metrical shape for different injection moulds. Their structuresand geometrical shapes are independent of the plastic mouldedproducts, but their sizes can be changed according to theplastic products. Therefore, it can be concluded that an injectionmould is actually a mechanical assembly that consists ofproduct-dependent parts and product-independent parts. Figure1 shows the assembly structure of an injection mould.The design of a product-dependent part is based on extractingthegeometryfrom theplasticproduct.Inrecentyears,CAD/CAM technology has been successfully used to helpmould designers to design the product-dependent parts. TheMouldMouldbaseCoolFillLayoutPlugSocketCav_1Cav_2CA-plateGuild-bushTCP-plateBep-plateCb-plateEa-plateEb-plateGuid-pinIp-plateRet-pinSliderbodyguideStop-blkHeel-blkheadCoreCavityProduct-independent partProduct-dependent partMove-halfFixed-halfFig. 1. Assembly structure of an injection mould.740X. G. Ye et al.automatic generation of the geometrical shape for a product-dependent part from the plastic product has also attracted alot of research interest 1,2. However, little work has beencarried out on the assembly modelling of injection moulds,although it is as important as the design of product-dependentparts. The mould industry is facing the following two difficult-ies when use a CAD system to design product-independentparts and the whole assembly of an injection mould. First,there are usually around one hundred product-independent partsin a mould set, and these parts are associated with each otherwith different kinds of constraints. It is time-consuming forthe designer to orient and position the components in anassembly. Secondly, while mould designers, most of the time,think on the level of real-world objects, such as screws, plates,and pins, the CAD system uses a totally different level ofgeometrical objects. As a result, high-level object-oriented ideashave to be translated to low-level CAD entities such as lines,surfaces, or solids. Therefore, it is necessary to develop anautomatic assembly modelling system for injection moulds tosolve these two problems. In this paper, we address the follow-ing two key issues for automatic assembly modelling: rep-resenting a product-independent part and a mould assembly ina computer; and determining the position and orientation of acomponent part in an assembly.This paper gives a brief review of related research inassembly modelling, and presents an integrated representationfor the injection mould assembly. A simplified geometric sym-bolic method is proposed to determine the position and orien-tation of a part in the mould assembly. An example of auto-matic assembly modelling of an injection mould is illustrated.2.Related ResearchAssembly modelling has been the subject of research in diversefields, such as, kinematics, AI, and geometric modelling. Lib-ardi et al. 3 compiled a research review of assembly model-ling. They reported that many researchers had used graphstructures to model assembly topology. In this graph scheme,the components are represented by nodes, and transformationmatrices are attached to arcs. However, the transformationmatrices are not coupled together, which seriously affects thetransformation procedure, i.e. if a subassembly is moved, allits constituent parts do not move correspondingly. Lee andGossard 4 developed a system that supported a hierarchicalassembly data structure containing more basic informationabout assemblies such as “mating feature” between the compo-nents. The transformation matrices are derived automaticallyfrom the associations of virtual links, but this hierarchicaltopology model represents only “part-of” relations effectively.Automatically inferring the configuration of components inan assembly means that designers can avoid specifying thetransformation matrices directly. Moreover, the position of acomponent will change whenever the size and position of itsreference component are modified. There exist three techniquesto infer the position and orientation of a component in theassembly: iterative numerical technique, symbolic algebraictechnique, and symbolic geometric technique. Lee and Gossard5 proposed an iterative numerical technique to compute thelocation and orientation of each component from the spatialrelationships. Their method consists of three steps: generationof the constraint equations, reducing the number of equations,and solving the equations. There are 16 equations for “against”condition, 18 equations for “fit” condition, 6 property equationsfor each matrix, and 2 additional equations for a rotationalpart. Usually the number of equations exceeds the number ofvariables, so a method must be devised to remove the redundantequations. The NewtonRaphson iteration algorithm is used tosolve the equations. This technique has two disadvantages:first, the solution is heavily dependent on the initial solution;secondly, the iterative numerical technique cannot distinguishbetween different roots in the solution space. Therefore, itis possible, in a purely spatial relationship problem, that amathematically valid, but physically unfeasible, solution canbe obtained.Ambler and Popplestone 6 suggested a method of comput-ing the required rotation and translation for each componentto satisfy the spatial relationships between the components inanassembly.Sixvariables(threetranslationsandthreerotations) for each component are solved to be consistent withthe spatial relationships. This method requires a vast amountof programming and computation to rewrite related equationsin a solvable format. Also, it does not guarantee a solutionevery time, especially when the equation cannot be rewrittenin solvable forms.Kramer 7 developed a symbolic geometric approach fordetermining the positions and orientations of rigid bodies thatsatisfy a set of geometric constraints. Reasoning about thegeometric bodies is performed symbolically by generating asequence of actions to satisfy each constraint incrementally,which results in the reduction of the objects available degreesof freedom (DOF). The fundamental reference entity used byKramer is called a “marker”, that is a point and two orthogonalaxes. Seven constraints (coincident, in-line, in-plane, parallelFz,offsetFz, offsetFx and helical) between markers are defined.For a problem involving a single object and constraints betweenmarkers on that body, and markers which have invariant attri-butes, action analysis 7 is used to obtain a solution. Actionanalysis decides the final configuration of a geometric object,step by step. At each step in solving the object configuration,degrees of freedom analysis decides what action will satisfyone of the bodys as yet unsatisfied constraints, given theavailable degrees of freedom. It then calculates how that actionfurther reduces the bodys degrees of freedom. At the end ofeach step, one appropriate action is added to the metaphoricalassembly plan. According to Shah and Rogers 8, Kramerswork represents the most significant development for assemblymodelling. This symbolic geometric approach can locate allsolutionsto constraintconditions, andis computationallyattractive compared to an iterative technique, but to implementthis method, a large amount of programming is required.Although many researchers have been actively involved inassembly modelling, little literature has been reported on fea-ture based assembly modelling for injection mould design.Kruth et al. 9 developed a design support system for aninjection mould. Their system supported the assembly designfor injection mouldsthrough high-level functionalmouldobjects (components and features). Because their system wasAutomated Assembly Modelling741based on AutoCAD, it could only accommodate wire-frameand simple solid models.3.Representation of Injection MouldAssembliesThe two key issues of automated assembly modelling forinjection moulds are, representing a mould assembly in com-puters, and determining the position and orientation of a pro-duct-independent part in the assembly. In this section, wepresent an object-oriented and feature-based representation forassemblies of injection moulds.The representation of assemblies in a computer involvesstructural and spatial relationships between individual parts.Such a representation must support the construction of anassembly from all the given parts, changes in the relativepositioning of parts, and manipulation of the assembly as awhole. Moreover, the representations of assemblies must meetthe following requirements from designers:1. It should be possible to have high-level objects ready touse while mould designers think on the level of real-world objects.2. The representation of assemblies should encapsulate oper-ational functions to automate routine processes such aspocketing and interference checks.To meet these requirements, a feature-based and object-orientedhierarchical model is proposed to represent injection moulds.An assembly may be divided into subassemblies, which in turnconsists of subassemblies and/or individual components. Thus,a hierarchical model is most appropriate for representing thestructural relations between components. A hierarchy impliesa definite assembly sequence. In addition, a hierarchical modelcan provide an explicit representation of the dependency ofthe position of one part on another.Feature-based design 10 allows designers to work at asomewhat higher level of abstraction than that possible withthe direct use of solid modellers. Geometric features areinstanced, sized, and located quickly by the user by specifyinga minimum set of parameters, while the feature modeller worksout the details. Also, it is easy to make design changes becauseof the associativities between geometric entities maintained inthe data structure of feature modellers. Without features,designers have to be concerned with all the details of geometricconstruction procedures required by solid modellers, and designchanges have to be strictly specified for every entity affectedby the change. Moreover, the feature-based representation willprovide high-level assembly objects for designers to use. Forexample, while mould designers think on the level of a real-world object, e.g. a counterbore hole, a feature object of acounterbore hole will be ready in the computer for use.Object-oriented modelling 11,12 is a new way of thinkingabout problems using models organised around real-world con-cepts. The fundamental entity is the object, which combinesboth data structures and behaviour in a single entity. Object-oriented models are useful for understanding problems anddesigning programs and databases. In addition, the object-oriented representation of assemblies makes it easy for a“child” object to inherit information from its “parent”.Figure 2 shows the feature-based and object-oriented hier-archical representation of an injection mould. The represen-tation is a hierarchical structure at multiple levels of abstraction,from low-level geometric entities (form feature) to high-levelsubassemblies. The items enclosed in the boxes represent“assembly objects” (SUBFAs, PARTs and FFs); the solid linesrepresent “part-of” relation; and the dashed lines representother relationships. Subassembly (SUBFA) consists of parts(PARTs). A part can be thought of as an “assembly” of formfeatures (FFs). The representation combines the strengths of afeature-based geometric model with those of object-orientedmodels. It not only contains the “part-of” relations betweenthe parent object and the child object, but also includes aricher set of structural relations and a group of operationalfunctions for assembly objects. In Section 3.1, there is furtherdiscussion on the definition of an assembly object, and detailedrelations between assembly objects are presented in Section Definition of Assembly ObjectsIn our work, an assembly object, O, is defined as a unique,identifiable entity in the following form:O = (Oid, A, M, R)(1)Where:Oid is a unique identifier of an assembly object (O).A is a set of three-tuples, (t, a, v). Each a is called anattribute of O, associated with each attribute is a type,t, and a value, v.M is a set of tuples, (m, tc1, tc2, %, tcn, tc). Eachelement of M is a function that uniquely identifies amethod. The symbol m represents a method name; andmethods define operations on objects. The symbol tci(iFig. 2. Feature-based, object-oriented hierarchical representation.742X. G. Ye et al.= 1, 2, %, n) specifies the argument type and tc specifiesthe returned value type.R is a set of relationships among O and other assemblyobjects.Therearesixtypesofbasicrelationshipsbetween assembly objects, i.e. Part-of, SR, SC, DOF,Lts, and Fit.Table 1 shows an assembly object of injection moulds, e.g.ejector. The ejector in Table 1 is formally specified as:(ejector-pinF1, (string, purpose, ejecting moulding),(string, material, nitride steel), (string, catalogFno,THX),(checkFinterference(), boolean), (pocketFplate(), boolean),(part-of ejectionFsys), (SR Align EBFplate), (DOF Tx,Ty).In this example, purpose, material and catalogFno areattributes with a data type of string; checkFinterference andpocketFplate are member functions; and Part-of, SR and DOFare relationships.3.2Assembly RelationshipsThere are six types of basic relationships between assemblyobjects, Part-of, SR, SC, DOF, Lts, and Fit.Part-ofAn assembly object belongs to its ancestor object.SRSpatial relations: explicitly specify the positionsandorientationsofassemblyobjectsinanassembly.Foracomponentpart,itsspatialrelationship is derived from spatial constraints(SC).SCSpatial constraints: implicitly locate a componentpart with respect to the other parts.DOFDegrees of freedom: are allowable translational/rotational directions of motion after assembly, withor without limits.LtsMotion limits: because of obstructions/interferences,the DOF may have unilateral or bilateral limits.FitSize constraint: is applied to dimensions, in orderto maintain a given class of fit.Table 1. Definition of an assembly object-ejector.Object Oidejector-pinF1Instance-ofEjectorFpinDerived from ejector classAPurpose “ejecting moulding” Type stringMaterial “nitrided steel”Type stringCatalogFno “THX”Type stringMCheckFinterferenceCheck interference(coolFobj)between ejectors andcooling linesPocketFplate()Make a hole on plate toaccommodate ejector pinsRPart-ofejectorFsysSRalign with EBplateDOFTx, TyAmong all the elements of an assembly object, the relation-ships are most important for assembly design. The relationshipsbetween assembly objects will not only determine the positionof objects in an assembly, but also maintain the associativitiesbetween assembly objects. In the following sub-sections, wewill illustrate the relationships at the same assembly level withthe help of examples.3.2.1Relationships Between Form FeaturesMould design, in essence, is a mental process; mould designersmost of the time think on the level of real-world objects suchas plates, screws, grooves, chamfers, and counter-bore holes.Therefore, it is necessary to build the geometric models of allproduct-independent parts from form features. The moulddesigner can easily change the size and shape of a part,because of the relations between form features maintained inthe part representation. Figure 3(a) shows a plate with acounter-bore hole. This part is defined by two form features,i.e. a block and a counter-bore hole. The counter-bore hole(FF2) is placed with reference to the block feature FF1, usingtheir local coordinates F2and F1, respectively. Equations (2)(5) show the spatial relationships between the counter-borehole (FF2) and the block feature (FF1). For form features,there is no spatial constraint between them, so the spatialrelationships are specified directly by the designer. The detailedassembly relationships between two form features are definedas follows:SR(FF2, FF1):F2i= F1i(2)F2j= F1j(3)Fig. 3. Assembly relationships.Automated Assembly Modelling743F2k= F1k(4)r2F= r1F+ b22*F1j+ AF1*F1i(5)DOF:ObjFhasF1FRDOF(FF2, F2j)The counter-bore feature can rotate about axis F2j.LTs(FF2, FF1):AF1, b11 0.5*b21(6)Fit (FF2, FF1):b22= b12(7)WhereF and r are the orientation and position vectors of fea-tures.F1= (F1i, F1j, F1k),F2= (F2i, F2j, F2k).bijis the dimension of form features, Subscript i isfeature number, j is dimension number.AF1is the dimension between form features.Equations (2)(7) present the relationships between the formfeature FF1and FF2. These relationships thus determine theposition and orientation of a form feature in the part. Takingthe part as an assembly, the form feature can be consideredas “components” of the assembly.The choice of form features is based on the shape character-istics of product-independent parts. Because the form featuresprovided by the Unigraphics CAD/CAM system 13 can meetthe shape requirements of parts for injection moulds and thespatial relationships between form features are also maintained,we choose them to build the required part models. In additionto the spatial relationships, we must record LTs, Fits relation-ships for form features, which are essential to check thevalidity of form features before updating the models in theCAD system.3.2.2Relationships Between PartsIn an assembly, the position and orientation of a part is usuallyassociated with another part. Figure 3(b) shows a plate (PP1)and a screw (PP2). The relative placement of the screw isconstrained by the counter-bore hole on the plate. The relation-ships between the screw and the plate are defined as follows:SR(PP2, PP1):P2= MpP1(8)r2= Mrr1(9)SC(PP2, PP1):mate(f1,f2) axisFalign(axisF1, axisF2)DOF:ObjFhasF1FRDOF(PP2, P2j)The screw can rotate about P2jof the plate.LTs(PP2, PP1):A22, A12(10)Fits(PP2, PP1):A13= A21+ cc(11)Where:P1and P2are the orientation vectors of the plate andthe screw, and P1= (P1i, P1j, P1k), P2= (P2i, P2j, P2k).Mpand Mrare the transformation matrix between thescrew and the plate. is a Boolean operator.mate and axisFalign are constraints (detailed discussionabout them are given in the next section).r is a position vector.Aijis dimension of parts. Subscript i is part number, andj is dimension number.cc is the clearance between the screw and the plate.As we can see in Eqs. (8) and (9), it is essential to calculatethe matrix Mpand Mrto determine the position and orientationof the screw with reference to the plate. Mpand Mrcan bederived from the spatial constraints (SC). This derivationrequires the task of inferring the configuration of a part in anassembly, which will be discussed in the next section.We have presented a representation of the injection mouldassembly in a computer. At this stage, it might be worthwhileto summarise the benefits of this representation. Assembliesare represented as a collection of subassemblies that in turnmay consist of subassemblies and/or component parts, and acomponent part can further be considered as the assembly ofform features. Such hierarchical relationships imply an orderingon the assembly sequence and a parentchild link. The feature-based representation not only allows designers to work at ahigh-level of abstraction while designing individual parts, butalso extends the feature paradigm to assembly modelling,because this representation allows a component to be changedparametrically with the other components consequently havingtheir positions changed accordingly. The object-oriented rep-resentation can combine both the data structure and operationin an object. The encapsulated operational functions in anassembly object can help to automate the routine processessuch as the pocketing and interference check.4.Inferring Part Configuration in theAssemblyAs we can see from Eqs (8) and (9), the positions andorientations of parts in an assembly are eventually representedby the transformation matrices. For the sake of convenience,the spatial relationships are usually specified by the high-levelmating conditions such as “mate”, “align” and “parallel”. Thus,it is essential to derive automatically the explicit transformationmatrices between parts from implicit constraint relationships.Three techniques to infer the configurations of parts in anassembly have been discussed in Section 2. Because the sym-bolic geometric approach can locate all solutions to constraintequations withpolynomial time complexity, we use thisapproach to determine the positions and orientations of partsin an assembly. To implement this approach in assemblymodelling software, a large amount of programming is required.Therefore, a simplified geometric approach is proposed todetermine the positions and orientations of parts in an assembly.744X. G. Ye et al.In the symbolic geometric approach, determining positionsand orientations of parts is performed symbolically by generat-ing a sequence of actions to satisfy each constraint incremen-tally. The information required to satisfy each constraintincrementally is stored in a table of “plan fragments”. Eachplan fragment is a procedure that specifies a sequence ofmeasurements and actions that move parts in such a way asto satisfy the corresponding constraint. The plan fragment alsorecords the objects new degrees of freedom (DOFs) andassociated geometric invariants. Conceptually, Kramers planfragment table is a 3D dispatch table. We use TDOF torepresent translational degrees of freedom and RDOF to rep-resent rotational degrees of freedom. Then an entry in the planfragment table has the following form:planFfragment: kTDOF, RDOF, constraintFtypelTDOF = 0, 1, 2, 3RDOF = 0, 1, 2, 3constraintFtype=coincident,in-line,in-plane,parallelFz, offsetFz, offsetFx and helicalThe plan fragment table is an exhaustive enumeration of allthe states in the search space for the problem of moving anobject to satisfy a series of constraints between markers onthe object and markers fixed in the global coordinate frame.To enumerate the combination of different values of the abovethree parameters, 82 entries will be generated 7. If the searchspace for the problem can be reduced the number of entriesin a plan fragment table will decrease. To achieve this, thenumber of enumerate values for entry parameters must bedecreased. For example, for a specified constraint type, if theenumeration values of TDOF change from 0,1,2,3 to 0,3,then the search space is reduced.After a careful analysis of the constraints between compo-nents of an injection mould, four basic primitive constraints areintroduced: in-line, parallelFz, parallelFz1 and parallelFoffset.Their definitions and algebraic equations are as follows:in-line(M1, M2): M1lies on the line through M2parallel to theZ-axis of M2.ugmp(M1) gmp(M2)u gmz(M2)u = 0(12)parallelFz(M1, M2): the Z-axes of markers M1and M2areparallel and have the same direction.gmz(M1) gmz(M2) = 1(13)parallelFz1(M1, M2): the Z-axes of markers M1and M2areparallel and have the opposite direction.gmz(M1) gmz(M2) = 1(14)paralleFoffset(M1, M2, d): Applicable only in conjunctionwith parallelFz or parallelFz1, specifies the distancebetween M1position and M2position.gmp(M1) gmp(M2) = d(15)Where:M1and M2are markers.gmp(M) is the global marker position.gmz(M) is the global marker Z-axis.gmx(M) the global marker X-axis.d is the distance between M1and M2.In our simplified symbolic geometric approach, the enumer-ation vales of constraint types are inFline, parallelFz, par-relFz1, paralleFoffset. Compared with Kramers symbolic geo-metric approach, our constraint types are reduced from sevento four. This simplification will reduce the number of entriesin the plan fragment table. Based on these four primitiveconstraints, three high-level constraints were synthesised fortheusersconvenience.Theyaremate(M1,M2,d),planeFalign(M1, M2, d), and axisFalign(M1, M2). Their defi-nitions are given as follows:mate(M1, M2, d):parallelFz1(M1, M2) parallelFoffset(M1, M2, d)planeFalign(M1, M2, d):parallelFz(M1, M2) parallelFoffset(M1, M2, d)axisFalign(M1, M2):parallelFz(M1, M2) inFline(M1, M2)The assembly objects in an injection mould can have one,two or three synthesised constraints. For two and three syn-thesised constraints, the constraint sequence is further restricted.The sequences are as follows:mate(M1, M2, d) planeFalign(M3, M4, d2)mate(M1, M2, d1) axisFalign(M3, M4)planeFalign(M1, M2, d) axisFalign(M3, M4)mate(M1,M2,d1)axisFalign(M3,M4,d2)axisFalign(M5, M6).Because of these restrictions on the constraint sequences,the number of entries in our plan fragment table is substantiallyreduced. To solve for one, two or three constraints allowed inour system, only nine entries are required. For interactiveaddition of components to the assembly, more constraint typesand free sequences will increase the flexibility for users. How-ever, in automatic assembly modelling for an injection mould,as the spatial relationships are predefined in assembly objects,some of the sequence restrictions do not matter. With theabove-defined synthesised constraints, the structural relation-ships of a component part can be specified in the database ofthe components. When adding a component part to the mouldassembly, the system will first decompose the synthesisedconstraints into primitive constraints, then generate a groupof fragment plans to orient and position the component inthe assembly.5.Automated Assembly Modelling ofInjection MouldsAnyassemblyofinjectionmouldsconsistsofproduct-independent parts and product-dependent parts. The design ofindividual product-dependent parts is based on the geometryof the plastic part 1,2. Usually the product-dependent partshave the same orientation as that of the top-level assembly,and their positions are specified directly by the designer. Asfor the design of product-independent parts, conventionally,mould designers select the structures from the catalogues,Automated Assembly Modelling745build the geometric models for selected structures of product-independent parts, and then add the product-independent partsto the assembly of the injection mould. This design process istime-consuming and error-prone. In our system, a database isbuiltforallproduct-independentpartsaccordingtotheassembly representation and object definition described in Sec-tion 3. This database not only contains the geometric shapesand sizes of the product-independent parts, but also includesthe spatial constraints between them. Moreover, some routinefunctions such as interference check and pocketing are encapsu-lated in the database. Therefore, the mould designer must selectthe structure types of product-independent parts from the userinterfaces, and then the software will automatically calculatethe orientation and position matrices for these parts, and addthem to the assembly.5.1Mould Base SubassemblyAs can be seen from Fig. 1, the product-independent parts canbe further classified as the mould base and standard parts. Amould base is the assembly of a group of plates, pins, guidebushes, etc. Besides shaping the product, a mould has to fulfila number of functions such as clamping the mould, leadingand aligning the mould halves, cooling, ejecting the product,etc. Most moulds have to incorporate the same functionality,which results in a similarity of the structural build-up. Someform of standardisation in mould construction has been adopted.A mould base is the result of this standardisation.According to the feature-based and object-oriented assemblyrepresentation, the feature-based solid models for componentparts of the mould base are first constructed; next, the assemblyobjectsaredefinedbyestablishingrelationshipsbetweencomponents and encapsulating some functions in the componentparts; then, using these assembly objects, a hierarchical subas-sembly object a mould base can be formed. This mouldbase object can be instantiated by a group of data from thecatalogue database. Figure 4 shows the instantiation of themould base object to generate the specified mould base. Thisspecified mould base instance can be added automatically tothe mould assembly. The structural relations between the mouldbase subassembly and top assembly can be expressed by Eqs.(8) and (9), where Mpand Mrare the unit matrices.5.2Automatic Addition of Standard PartsA standard part is an assembly object. It can be definedaccording to Eq. (1) in Section 3.1. In the database, the spatialconstraints are specified by mate, planeFalign and axisFalign,but unlike the mould base, the position and orientation matricesof a standard part are left unknown. During instantiation,the software then automatically infers the explicit structuralrelationshipsbyusingthesimplifiedsymbolicgeometricapproach described in Section 4.5.3Pocketing for Assembly ObjectsOne of the important issues for automatic assembly design isthe automation of the pocketing process. Pocketing is anFig. 4. Instantiation of mould base.operation that makes an empty space in corresponding compo-nents to accommodate the inserted components. When an ejec-tor is added to the assembly, an empty space is required onthe EA plate to accommodate the ejector, as shown in Fig. 5.Since an object-oriented representation is adopted, eachassembly object can be represented by two solids, the realobject and the virtual object. The virtual object is modelledaccording to the space that a real object will occupy. Wheneveran assembly object is added to an assembly, its virtual objectisalsoaddedtotheassembly.TheoperationfunctionpocketFplate() in M of O will subtract the virtual object fromthe corresponding components (see Eq. (1) and Table 1).Moreover, because there are associativities between the virtualobject and real object, the pockets on the corresponding compo-nents will change with the modification of the real object.Fig. 5. Pocketing of assembled objects.746X. G. Ye et al.This automatic pocketing function further demonstrates theadvantage of an object-oriented representation.6.System ImplementationBased on the Unigraphics system 13, the proposed feature-based and object-oriented assembly scheme and automation ofassembly modelling have been implemented in the IMOLDsystem 14 developed at the National University of Singapore.The Unigraphics sys