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注塑模具自動裝配造型 X. G. Ye, J. Y. H. Fuh and K. S. Lee 機械和生產(chǎn)工程部，新加坡國立大學，新加坡 注射模是一種由與塑料制品有關(guān)的和與制品無關(guān)的零部件兩大部分組成的機械裝置。本文提出了（有關(guān)）注射模裝配造型的兩個主要觀點，即描述了在計算機上進行注射模裝配以及確定裝配中與制品無關(guān)的零部件的方向和位置的方法，提出了一個基于特征和面向?qū)ο蟮谋磉_式以描述注射模等級裝配關(guān)系，該論述要求并允許設(shè)計者除了考慮零部件的外觀形狀和位置外，還要明確知道什么部份最重要和為什么。因此，它為設(shè)計者進行裝配設(shè)計（DFA）提供了一個機會。同樣地，為了根據(jù)裝配狀態(tài)推斷出裝配體中裝配對象的結(jié)構(gòu)，一種簡化的特征幾何學方法也誕生了。在提出的表達式和簡化特征幾何學的基礎(chǔ)上，進一步深入探討了自動裝配造型的方法。 關(guān)鍵字：裝配造型；基于特征；注射模；面向?qū)ο蟆? 1、簡介 注射成型是生產(chǎn)塑料模具產(chǎn)品最重要的工藝。需要用到的兩種裝備是：注射成型機和注射?！，F(xiàn)在常用的注射成型機即所謂的通用機，在一定尺寸范圍內(nèi)，可以用于不同形狀的各種塑料模型中，但注射模的設(shè)計就必須隨塑料制品的變化而變化。模型的幾何因素不同，它們的構(gòu)造也就不同。注射模的主要任務(wù)是把塑料熔體制成塑料制品的最終形狀，這個過程是由型芯、型腔、鑲件、滑塊等與塑料制品有關(guān)的零部件完成的，它們是直接構(gòu)成塑料件形狀及尺寸的各種零件，因此，這些零件稱為成型零件。（在下文，制品指塑料模具制品，部件指注射模的零部件。）除了注射成型外，注射模還必須完成分配熔體、冷卻、開模、傳輸、引導(dǎo)運動等任務(wù)，而完成這些任務(wù)的注射模組件在結(jié)構(gòu)和形狀上往往都是相似的，它們的結(jié)構(gòu)和形狀并不取決于塑料模具，而是取決于塑料制品。圖1顯示了注射模的結(jié)構(gòu)組成。 圖1 注射模的結(jié)構(gòu) 成型零件的設(shè)計從塑料制品中分離了出來。近幾年，CAD/CAM技術(shù)已經(jīng)成功的應(yīng)用到成型零件的設(shè)計上。成型零件的形狀的自動化生成也引起了很多研究者的興趣，不過很少有人在其上付諸實踐，雖然它也象結(jié)構(gòu)零件一樣重要。現(xiàn)在，模具工業(yè)在應(yīng)用計算機輔助設(shè)計系統(tǒng)設(shè)計成型零件和注射成型機時，遇到了兩個主要困難。第一，在一個模具裝置中，通常都包括有一百多個成型零部件，而這些零部件又相互聯(lián)系，相互限制。對于設(shè)計者來說，確定好這些零部件的正確位置是很費時間的。第二，在很多時候，模具設(shè)計者已想象出工件的真實形狀，例如螺絲，轉(zhuǎn)盤和銷釘，但是CAD系統(tǒng)只能用于另一種信息的操作。這就需要設(shè)計者將他們的想法轉(zhuǎn)化成CAD系統(tǒng)能接受的信息(例如線，面或者實體等)。因此，為了解決這兩個問題，很有必要發(fā)展一種用于注射模的自動裝配成型系統(tǒng)。在此篇文章里，主要講述了兩個觀點：即成型零部件和模具在計算機上的防真裝配以及確定零部件在模具中的結(jié)構(gòu)和位置。 這篇文章概括了關(guān)于注塑成型的相關(guān)研究，并對注射成型機有一個完整的闡述。通過舉例一個注射模的自動裝配造型，提出一種簡化的幾何學符號法，用于確定注射模具零部件的結(jié)構(gòu)和位置。 2.相關(guān)研究 在各種領(lǐng)域的研究中，裝配造型已成為一門學科，就像運動學、人工智能學、模擬幾何學一樣。Libardi作了一個關(guān)于裝配造型的調(diào)查。據(jù)稱，很多研究人員已經(jīng)開始用圖表分析模型會議拓撲。在這個圖里，各個元件由節(jié)點組成的，再將這些點依次連接成線段。然而這些變化矩陣并沒有緊緊的連在一起，這將嚴重影響整體的結(jié)構(gòu)，即，當其中某一部分移動了，其他部分并不能做出相應(yīng)的移動。Lee and Gossard開發(fā)了一種新的系統(tǒng)，支持包含更多的關(guān)于零部件的基本信息的一種分級的裝配數(shù)據(jù)結(jié)構(gòu)，就像在各元件間的“裝配特征”。變化矩陣自動從實際的線段間的聯(lián)系得到，但是這個分級的拓撲模型只能有效地代表“部分”的關(guān)系。 自動判別裝配組件的結(jié)構(gòu)意味著設(shè)計者可避免直接指定變化的矩陣，而且，當它的參考零部件的尺寸和位置被修改的時候，它的位置也將隨之改變?，F(xiàn)在有三種技術(shù)可以推斷組件在模具中的位置和結(jié)構(gòu)：反復(fù)數(shù)值技術(shù),象征代數(shù)學技術(shù)，以及象征幾何學技術(shù)。Lee and Gossard提出一項從空間關(guān)系計算每個組成元件的位置和方向的反復(fù)數(shù)值技術(shù)。他們的理論由三步組成：產(chǎn)生條件方程式，降低方程式數(shù)量，解答方程式。方程式有：16個滿足未知條件的方程式，18個滿足已知條件的方程式，6個滿足各個矩陣的方程式以及另外的兩個滿足旋轉(zhuǎn)元件的方程式。通常方程式的數(shù)量超過變量的數(shù)量時，應(yīng)該想辦法去除多余的方程式。牛頓迭代法常用來解決這種方程式。不過這種方法存在兩種缺點：第一，它太依賴初始解；第二：反復(fù)的數(shù)值技術(shù)在解決空間內(nèi)不能分清不同的根。因此，在一個完全的空間關(guān)系問題上，有可能解出來的結(jié)果在數(shù)學理論上有效，但實際上卻是行不通的。 Ambler和Popplestone提議分別計算每個零部件的旋轉(zhuǎn)量和轉(zhuǎn)變量以確定它們之間的空間關(guān)系，而解出的每個零部件的6個變量(3個轉(zhuǎn)變量和3旋轉(zhuǎn)量)要和它們的空間關(guān)系一致。這種方法要求大量的編程和計算，才能用可解的形式重寫有關(guān)的方程式。此外，它不能保證每次都能求出結(jié)果，特別是當方程式不能被以可解答的形式重寫時。 為了能確定出滿足一套幾何學限制條件的剛體的位置與方向，Kramer開發(fā)了一種特征幾何學方法。通過產(chǎn)生一連串滿足逐漸增長的限制條件的動作推斷其幾何特征，這樣將減少物體的自由度數(shù)。Kramer使用的基本參考實體稱為一個"標識"，由一個點和兩正交軸構(gòu)成。標識間的7個限制條件（coincident, in-line, in-plane, parallelFz,offsetFz, offsetFx and helical)都被定了義。對于一個包括獨立元件、相互約束的標識和不變的標識的問題來說，可以用動作分析法來解決問題，它將一步一步地最后求出物體的最終的幾何構(gòu)造。在確定物體構(gòu)造的每一個階段，自由度分析將決定什么動作能提供滿足限制物體未加限制部位的自由度。然后計算該動作怎樣能進一步降低物體的自由度數(shù)。在每個階段的最后，給隱喻的裝配計劃加上合適的一步。根據(jù)Shah和Rogers的分析，Kramer的理論代表了注射模具最顯著的發(fā)展，他的特征幾何學方法能解出全部的限制條件。和反復(fù)的數(shù)值技術(shù)相比，他的這種方法更具吸引力。不過要實行這種方法，需要大量的編程。 現(xiàn)在雖然已有很多研究者開始研究注射成型機，但仍很少有學者將注意力放在注射模設(shè)計上。Kruth開發(fā)了一個注射模的設(shè)計支援系統(tǒng)。這個系統(tǒng)通過高級的模具對象(零部件和特征)支持注射模的成型設(shè)計。因為系統(tǒng)是在AUTOCAD的基礎(chǔ)上設(shè)計的，因此它只適于線和簡單的實體模型操作。 3.注射模裝配概述 主要講述了關(guān)于注射模自動裝配造型的兩個方面：注射模在電腦上的防真裝配和確定結(jié)構(gòu)零件在裝配中的位置和方向。在這個部分，我們基于特征和面向?qū)ο笳撌隽俗⑸淠Ｑb配。 注射模在電腦上的防真裝配包含著注射模零部件在結(jié)構(gòu)上和空間上的聯(lián)系。這種防真必須支持所有給定零部件的裝配、在相互關(guān)聯(lián)的零部件間進行變動以及整體上的操作。而且防真裝配也必須滿足設(shè)計者的下列要求： 1． 支持能表達出模具設(shè)計者實體造型想象的高級對象。 2． 成型防真應(yīng)該有象現(xiàn)實一樣的操作功能，就如裝入和干擾檢查。 為了滿足這些要求，可用一個基于特征和面向?qū)ο蟮姆旨壞Ｐ蛠泶孀⑸淠！＿@樣便將模型分成許多部分，反過來由多段模型和獨立部分組成。因此，一個分級的模型最適合于描述各組成部分之間的結(jié)構(gòu)關(guān)系。一級表明一個裝配順序，另外，一個分級的模型還能說明一個部分相對于另一個部分的確定位置。 與直觀的固體模型操作相比，面向特征設(shè)計允許設(shè)計者在抽象上進行操作。它可以通過一最小套參數(shù)快速列出模型的特征、尺寸以及其方位。此外，由于特征模型的數(shù)據(jù)結(jié)構(gòu)在幾何實體上的聯(lián)系，設(shè)計者更容易更改設(shè)計。如果沒有這些特征，設(shè)計者在構(gòu)造固體模型幾何特征時就必須考慮到所有需要的細節(jié)。而且面向特征的防真為設(shè)計者提供了更高級的成型對象。例如，模具設(shè)計者想象出一個澆口的實體形狀，電腦就能將這個澆口造型出來。 面向?qū)ο笤煨头ㄊ且环N參照實物的概念去設(shè)計模型的新思維方式?；镜膱D素是能夠?qū)?shù)據(jù)庫和單一圖素的動作聯(lián)系起來的對象。面向?qū)ο蟮脑煨蛯斫鈫栴}并且設(shè)計程序和數(shù)據(jù)庫是很有用的。此外，面向?qū)ο蟮难b配體呈現(xiàn)方式使得“子”對象能繼承其“父”對象的信息變得更容易。 圖形2說明以特性為基礎(chǔ)和面向?qū)ο蟮姆謱拥谋硎疽环N插入模具。 表示是多重水平的提取的一種分層的結(jié)構(gòu)，從低水平的幾何學的實體(形成特性)到高水平的組件。 在盒子中被封入的項目代表“裝配對象”； 固體線代表“部分”關(guān)系； 同時，猛沖的線代表其它關(guān)系。 組件( SUBFA )包括部分( PART )。 一部分能被認為是形式特性( FF )的一種“裝配”。 表示把一個以特性為基礎(chǔ)的幾何學的模型的力與面向?qū)ο蟮哪Ｐ偷哪切┫嘟Y(jié)合。 它不僅包含父對象和子對象之間的“部分”關(guān)系，也包括富有的套結(jié)構(gòu)的關(guān)系和裝配對象的一群操作的功能。 在段中3.1，在裝配對象之間有有關(guān)一種裝配對象的定義的較進一步的討論，而詳盡的關(guān)系在3.2段中被提出。 3.1裝配對象的定義 在我們的工作中，一種裝配對象，O，以如下形式被定義為一個唯一而可辨認的實體： O = ( Oid，A，M，R ) ( 1 ) 在此式中： Oid是一種裝配對象( O )的一個唯一的標識符。 A是一套三元組，( t，a，v )。 每一元素a被稱為O的一種屬性，與每一屬性有關(guān)是一類型，t，和一種價值，v。 M是一套元組，( m，tc1，tc2，%，tcn，tc)。 M中每一個元素都有唯一識別方法。 符號m代表一種方法名稱； 同時，方法定義有關(guān)對象的操作。 符號tc (i= 1，2，%，n )規(guī)定爭論類型和符號tc退回的價值類型。 3.2形式特性之間的關(guān)系 模具設(shè)計在本質(zhì)中是一個智力的過程； 模具設(shè)計者大多數(shù)時間在真實客觀的對象諸如金屬板，螺絲釘，槽，斜面，和孔等思索設(shè)想。因此，用形式特性建設(shè)所有產(chǎn)品獨立部分的幾何學的模型是必要。 模具設(shè)計者能容易地改變一部分的大小和形狀，因為形式特性之間的關(guān)系保持在部分表示中。 圖形3(a )顯示一個金屬板帶有一個含有公差等級要求的孔。 這部分被兩個形式特性定義，即一個塊和含有公差等級要求的孔。 關(guān)于塊特性計數(shù)器開掘洞( FF2 )被放置FF1，使用他們本地分別地協(xié)調(diào)F2和F1，。 方程( 2)– ( 5 )顯示計數(shù)器開掘洞( FF2 )和塊特性( FF1 )之間的空間的關(guān)系。 對于形式特性，沒有他們之間的空間的約束，因此空間的關(guān)系被設(shè)計者直接指定。 兩形式特性之間的詳盡的裝配關(guān)系被定義如下： 4.在裝配中推斷部分配置 一種裝配中的若干部分的位置和方向最后通過轉(zhuǎn)換矩陣來表達。為了方便的緣故，空間的關(guān)系通常被諸如“伙伴”，“結(jié)盟”和“平行”的高水平的鋪席子的條件指定。 這樣，從含蓄的約束關(guān)系自動地引出若干部分之間的清晰明確的轉(zhuǎn)換矩陣是十分重要。推斷一種裝配中的若干部分的配置三種技術(shù)在段2.中已被討論了因為象征性幾何學的接近能以多項式時間復(fù)雜性定位所有關(guān)于約束方程的解決方案，我們使用這接近來確定位置和一種裝配中的若干部分的方向。 為了在裝配模擬軟件中執(zhí)行這接近，大量的編寫程序被要求。因此，一種簡化的幾何學的接近被建議確定位置和一種裝配中的若干部分的方向。 在象征性幾何學的接近中，確定位置和若干部分的方向被產(chǎn)生一系列行動執(zhí)行符號滿足每一逐漸增長的約束。被要求來滿足每一逐漸增長的約束的信息儲存在“計劃片段”的一個表格中。 每一計劃片段是規(guī)定一系列測量方法和行動的一個過程按照這樣一種方式移動部分對于滿足相應(yīng)的約束。 計劃片段也記錄新的自由度和聯(lián)系不變量的幾何不變式。 由于這些限制約束序列，我們的計劃片段桌子中的輸入的數(shù)字基本上被減少。 為了為了一，兩或者三個約束解決在我們的系統(tǒng)中允許，九種輸入僅僅被要求。 為了交互式的增加組成部分裝配，更多約束類型和自由的序列將為了用戶增加靈活性。 然而，在為了一種插入模具模擬的自動裝配中，當空間的關(guān)系被預(yù)先規(guī)定在裝配對象中時，一些序列限制不有關(guān)系。 有了上述的定義的合成約束，一個組成部分部分的結(jié)構(gòu)的關(guān)系能指定在組成部分的數(shù)據(jù)庫中。 當把一個組成部分部分添加到模具裝配時，系統(tǒng)將首先分解進入原始的約束的合成約束，然后產(chǎn)生一群片段計劃將組成部分指明方向并且定位在裝配中。 5.注射模的自動裝配 任何注射模具的裝配都由產(chǎn)品的局部和整體兩部分組成。產(chǎn)品的局部依賴產(chǎn)品的整體設(shè)計基于塑料的部分[ 1,2 ]的幾何學。 產(chǎn)品依賴部分通常有與那個同樣的方向頂端水平裝配，而他們的位置被設(shè)計者直接指定。 對于產(chǎn)品獨立部分的設(shè)計，常規(guī)，模具設(shè)計者從目錄中選擇結(jié)構(gòu)，為了產(chǎn)品若干部分的選擇的結(jié)構(gòu)建設(shè)幾何學的模型，而然后把產(chǎn)品獨立部分添加到插入模具的裝配。 這設(shè)計過程是時間消耗的和差錯容易傾向于。 在我們的系統(tǒng)中，一個數(shù)據(jù)庫為了所有產(chǎn)品獨立部分根據(jù)裝配表示被建造，而對象定義在段3.中不僅描述這數(shù)據(jù)庫包含產(chǎn)品獨立部分的幾何學的形狀和大小，也包括他們之間的空間的約束。 此外，一些日常事務(wù)發(fā)揮作用諸如干擾檢查和裝在衣袋內(nèi)被封裝在數(shù)據(jù)庫中。 因此，模具設(shè)計者必須從用戶接口中選擇產(chǎn)品獨立部分的結(jié)構(gòu)類型，而然后軟件將為了這些部分自動地計算方向和位置矩陣，而把他們添加到裝配。 5.1模具基礎(chǔ)組件 正如圖1所示，產(chǎn)品的獨立部分可以更進一步被分為摸具基礎(chǔ)和標準部分。摸具基礎(chǔ)是由一群金屬板，插腳，導(dǎo)套等等組成的。除了塑型產(chǎn)品，模具必須具有一系列功能，諸如，箝位，校準，冷卻，注塑等等。大多數(shù)產(chǎn)品不得不合并相同的功能，這導(dǎo)致了相似結(jié)構(gòu)的樹立。一些模具建筑形成的標準已經(jīng)被采用了。模具基礎(chǔ)起因于這個標準。 根據(jù)以特性為基礎(chǔ)和面向?qū)ο蟮难b配表示，模具基礎(chǔ)組成部分的以特性為基礎(chǔ)的固體模具首先被建造；其次，裝配對象被定義為在成分和壓縮功能一部分功能在組成零件之間建立關(guān)系；然后，利用這些組裝對象，一個分層的組裝對象——模具基礎(chǔ)——能被形成。這些模具基礎(chǔ)對象能通過目錄數(shù)據(jù)庫被例示。表4列出了模具基礎(chǔ)對象來產(chǎn)生指定的模具基礎(chǔ)的例子。這個指定的模具基礎(chǔ)實例能自動地添加到模具裝配。模具基礎(chǔ)部件和最高裝配的結(jié)構(gòu)關(guān)系能通過Eqs被表達。Mp和Mr所在的（8）和（9）式是單元矩陣。 5.2 標準零件的自動增加 一個標準零件是一個組裝對象。它可以通過章節(jié)3.1的公式（1）來定義。在數(shù)據(jù)庫中，空間約束用 mate，平面aling和軸align，而不像模具基礎(chǔ)，標準件的位置和方向的矩陣是未知的。在示例中，軟件通過利用單一的符號幾何來自動推斷章節(jié)4中描述的結(jié)構(gòu)關(guān)系。 5.3 裝配對象的包裝 自動裝配設(shè)計的一個重要問題是自動包裝過程。包裝是一個在相應(yīng)組成部分提供附著成分的真空區(qū)的操作。當一個驅(qū)動者被添加到裝配時，一個空的空間被要求在EA盤上調(diào)節(jié)驅(qū)動者，如表5所示。 由于面向?qū)ο蟮谋硎痉ū徊扇?，每一個裝配對象能被描述為兩個實體，實物和虛擬物。虛擬物通過被實物占據(jù)的空間模仿。只要一個裝配對象被添加到裝配中，它的虛擬對象也被添加到裝配中。操作發(fā)揮作用中的pocketFplate( ) M O將從相應(yīng)的組成部分(參看公式（1）和表1)。此外，因為在相應(yīng)的組成部分上在虛擬對象和真正的對象之間有聯(lián)系，包裝將隨真正的對象的修正而變化。 這種自動包裝功能更進一步顯示了面向?qū)ο蟊硎痉ǖ膬?yōu)勢。 6.基于Unigraphics系統(tǒng)[ 13 ]，所提出的以特性為基礎(chǔ)和面向?qū)ο蟮难b配計劃和自動化裝配模擬的系統(tǒng)在新加坡的國立大學被開發(fā)的IMOLD系統(tǒng)[ 14 ]中已被執(zhí)行。UG系統(tǒng)提供了一個友好的用戶應(yīng)用程序接口。通過這個接口，用戶可以調(diào)用UG的內(nèi)部功能，諸如增加裝配部件，修正參數(shù)等等。 圖6顯示的是一個注塑模具產(chǎn)品，這個產(chǎn)品的注塑模具組裝設(shè)計顯示在圖7（a）。固定一半組件的相應(yīng)的父子關(guān)系圖顯示在圖7（b）。裝配是由IMOLD系統(tǒng)設(shè)計。每一個模具基礎(chǔ)的零件都在裝配中自動定位。Unigraphics系統(tǒng)提供一個用戶友好應(yīng)用編寫程序接口(應(yīng)用程序接口)。 通過這接口，雖然Unigraphics為了給條件鋪席子提供功能，用戶能呼叫諸如把部分添加到一種裝配的Unigraphics內(nèi)部的功能，修改參數(shù)等等，所提出的接近仍然被需要推斷組成部分配置，因為在組成部分能被添加到裝配之前，計算自由的度是必要，而檢查給條件鋪席子的有效性。 圖6個展覽一種插入鑄造產(chǎn)品，因為圖被領(lǐng)進來，和設(shè)計的插入模具裝配這產(chǎn)品7(a )。 固定一半組件的相應(yīng)的“父與子”關(guān)系被領(lǐng)進來圖7(b )。 這裝配被系統(tǒng)設(shè)計。 每一模具基礎(chǔ)的盤子自動地被定位在裝配中。 諸如定位的圓環(huán)和驅(qū)逐者的標準的部分自動地被添加到裝配，因為這些標準部分也自動地被建立，和口袋。 7.結(jié)論 注射模具裝配以所提出的特性為基礎(chǔ)和面向?qū)ο蟮姆謱拥谋硎静粌H把特性范例擴展到裝配，由于擴展特性范例而給條件，插入和方向限制等等鋪席子到裝配設(shè)計設(shè)計，而且是封裝操作的功能和幾何學的約束，諸如自由的程度，諸如集合的組成部分的模糊變化修正甚至能在完成裝配過程之后被制定。 裝配對象的封裝有如下兩種優(yōu)勢： 首先，因為裝配的條件被封裝在裝配對象中，自動裝配設(shè)計容易執(zhí)行； 其次，對象裝配的封裝操作的功能使諸如裝在衣袋內(nèi)與干擾檢查的裝配設(shè)計的日常事務(wù)過程自動化。 所提出的簡單化的動作分析能基本上減少為了自動檢測校對模具裝配之內(nèi)組成部分干擾所需要的規(guī)劃設(shè)計的努力。 Int J Adv Manuf Technol (2000) 16:739747 2000 Springer-Verlag London Limited Automated Assembly Modelling for Plastic Injection Moulds X. G. Ye, J. Y. H. Fuh and K. S. Lee Department of Mechanical and Production Engineering, National University of Singapore, Singapore An injection mould is a mechanical assembly that consists of product-dependent parts and product-independent parts. This paper addresses the two key issues of assembly modelling for injection moulds, namely, representing an injection mould assembly in a computer and determining the position and orientation of a product-independent part in an assembly. A feature-based and object-oriented representation is proposed to represent the hierarchical assembly of injection moulds. This representation requires and permits a designer to think beyond the mere shape of a part and state explicitly what portions of a part are important and why. Thus, it provides an opportunity for designers to design for assembly (DFA). A simplified symbolic geometric approach is also presented to infer the configurations of assembly objects in an assembly according to the mating conditions. Based on the proposed representation and the simplified symbolic geometric approach, automatic assembly modelling is further discussed. Keywords: Assembly modelling; Feature-based; Injection moulds; Object-oriented 1. Introduction Injection moulding is the most important process for manufac- turing plastic moulded products. The necessary equipment con- sists of two main elements, the injection moulding machine and the injection mould. The injection moulding machines used today are so-called universal machines, onto which various moulds for plastic parts with different geometries can be mounted, within certain dimension limits, but the injection mould design has to change with plastic products. For different moulding geometries, different mould configurations are usually necessary. The primary task of an injection mould is to shape the 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 shapes Correspondence and offprint requests to: Dr Jerry Y. H. Fuh, Depart- ment of Mechanical and Production Engineering, National University of Singapore (NUS), 10 Kent Ridge Crescent, Singapore 119260. E-mail: mpefuhyhKnus.edu.sg and sizes of a cavity system are determined directly by the plastic moulded product, so all components of a cavity system are called product-dependent parts. (Hereinafter, product refers to a plastic moulded product, part refers to the component of an injection mould.) Besides the primary task of shaping the product, an injection mould has also to fulfil a number of tasks such as the distribution of melt, cooling the molten material, ejection of the moulded product, transmitting motion, guiding, and aligning the mould halves. The functional parts to fulfil these tasks are usually similar in structure and geo- metrical shape for different injection moulds. Their structures and geometrical shapes are independent of the plastic moulded products, but their sizes can be changed according to the plastic products. Therefore, it can be concluded that an injection mould is actually a mechanical assembly that consists of product-dependent parts and product-independent parts. Figure 1 shows the assembly structure of an injection mould. The design of a product-dependent part is based on extracting the geometry from the plastic product. In recent years, CAD/CAM technology has been successfully used to help mould designers to design the product-dependent parts. The Mould Mouldbase Cool Fill Layout Plug Socket Cav_1 Cav_2 CA-plate Guild-bush TCP-plate Bep-plate Cb-plate Ea-plate Eb-plate Guid-pin Ip-plate Ret-pin Slider body guide Stop-blk Heel-blk head Core Cavity Product-independent part Product-dependent part Move-half Fixed-half Fig. 1. Assembly structure of an injection mould. 740 X. G. Ye et al. automatic generation of the geometrical shape for a product- dependent part from the plastic product has also attracted a lot of research interest 1,2. However, little work has been carried out on the assembly modelling of injection moulds, although it is as important as the design of product-dependent parts. The mould industry is facing the following two difficult- ies when use a CAD system to design product-independent parts and the whole assembly of an injection mould. First, there are usually around one hundred product-independent parts in a mould set, and these parts are associated with each other with different kinds of constraints. It is time-consuming for the designer to orient and position the components in an assembly. 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 of geometrical objects. As a result, high-level object-oriented ideas have to be translated to low-level CAD entities such as lines, surfaces, or solids. Therefore, it is necessary to develop an automatic assembly modelling system for injection moulds to solve 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 in a computer; and determining the position and orientation of a component part in an assembly. This paper gives a brief review of related research in assembly modelling, and presents an integrated representation for 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 Research Assembly modelling has been the subject of research in diverse fields, 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 graph structures to model assembly topology. In this graph scheme, the components are represented by nodes, and transformation matrices are attached to arcs. However, the transformation matrices are not coupled together, which seriously affects the transformation procedure, i.e. if a subassembly is moved, all its constituent parts do not move correspondingly. Lee and Gossard 4 developed a system that supported a hierarchical assembly data structure containing more basic information about assemblies such as “mating feature” between the compo- nents. The transformation matrices are derived automatically from the associations of virtual links, but this hierarchical topology model represents only “part-of” relations effectively. Automatically inferring the configuration of components in an assembly means that designers can avoid specifying the transformation matrices directly. Moreover, the position of a component will change whenever the size and position of its reference component are modified. There exist three techniques to infer the position and orientation of a component in the assembly: iterative numerical technique, symbolic algebraic technique, and symbolic geometric technique. Lee and Gossard 5 proposed an iterative numerical technique to compute the location and orientation of each component from the spatial relationships. Their method consists of three steps: generation of 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 equations for each matrix, and 2 additional equations for a rotational part. Usually the number of equations exceeds the number of variables, so a method must be devised to remove the redundant equations. The NewtonRaphson iteration algorithm is used to solve the equations. This technique has two disadvantages: first, the solution is heavily dependent on the initial solution; secondly, the iterative numerical technique cannot distinguish between different roots in the solution space. Therefore, it is possible, in a purely spatial relationship problem, that a mathematically valid, but physically unfeasible, solution can be obtained. Ambler and Popplestone 6 suggested a method of comput- ing the required rotation and translation for each component to satisfy the spatial relationships between the components in an assembly. Six variables (three translations and three rotations) for each component are solved to be consistent with the spatial relationships. This method requires a vast amount of programming and computation to rewrite related equations in a solvable format. Also, it does not guarantee a solution every time, especially when the equation cannot be rewritten in solvable forms. Kramer 7 developed a symbolic geometric approach for determining the positions and orientations of rigid bodies that satisfy a set of geometric constraints. Reasoning about the geometric bodies is performed symbolically by generating a sequence of actions to satisfy each constraint incrementally, which results in the reduction of the objects available degrees of freedom (DOF). The fundamental reference entity used by Kramer is called a “marker”, that is a point and two orthogonal axes. 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 between markers on that body, and markers which have invariant attri- butes, action analysis 7 is used to obtain a solution. Action analysis 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 satisfy one of the bodys as yet unsatisfied constraints, given the available degrees of freedom. It then calculates how that action further reduces the bodys degrees of freedom. At the end of each step, one appropriate action is added to the metaphorical assembly plan. According to Shah and Rogers 8, Kramers work represents the most significant development for assembly modelling. This symbolic geometric approach can locate all solutions to constraint conditions, and is computationally attractive compared to an iterative technique, but to implement this method, a large amount of programming is required. Although many researchers have been actively involved in assembly 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 an injection mould. Their system supported the assembly design for injection moulds through high-level functional mould objects (components and features). Because their system was Automated Assembly Modelling 741 based on AutoCAD, it could only accommodate wire-frame and simple solid models. 3. Representation of Injection Mould Assemblies The two key issues of automated assembly modelling for injection 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, we present an object-oriented and feature-based representation for assemblies of injection moulds. The representation of assemblies in a computer involves structural and spatial relationships between individual parts. Such a representation must support the construction of an assembly from all the given parts, changes in the relative positioning of parts, and manipulation of the assembly as a whole. Moreover, the representations of assemblies must meet the following requirements from designers: 1. It should be possible to have high-level objects ready to use 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 as pocketing and interference checks. To meet these requirements, a feature-based and object-oriented hierarchical model is proposed to represent injection moulds. An assembly may be divided into subassemblies, which in turn consists of subassemblies and/or individual components. Thus, a hierarchical model is most appropriate for representing the structural relations between components. A hierarchy implies a definite assembly sequence. In addition, a hierarchical model can provide an explicit representation of the dependency of the position of one part on another. Feature-based design 10 allows designers to work at a somewhat higher level of abstraction than that possible with the direct use of solid modellers. Geometric features are instanced, sized, and located quickly by the user by specifying a minimum set of parameters, while the feature modeller works out the details. Also, it is easy to make design changes because of the associativities between geometric entities maintained in the data structure of feature modellers. Without features, designers have to be concerned with all the details of geometric construction procedures required by solid modellers, and design changes have to be strictly specified for every entity affected by the change. Moreover, the feature-based representation will provide high-level assembly objects for designers to use. For example, while mould designers think on the level of a real- world object, e.g. a counterbore hole, a feature object of a counterbore hole will be ready in the computer for use. Object-oriented modelling 11,12 is a new way of thinking about problems using models organised around real-world con- cepts. The fundamental entity is the object, which combines both data structures and behaviour in a single entity. Object- oriented models are useful for understanding problems and designing 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-level subassemblies. The items enclosed in the boxes represent “assembly objects” (SUBFAs, PARTs and FFs); the solid lines represent “part-of” relation; and the dashed lines represent other relationships. Subassembly (SUBFA) consists of parts (PARTs). A part can be thought of as an “assembly” of form features (FFs). The representation combines the strengths of a feature-based geometric model with those of object-oriented models. It not only contains the “part-of” relations between the parent object and the child object, but also includes a richer set of structural relations and a group of operational functions for assembly objects. In Section 3.1, there is further discussion on the definition of an assembly object, and detailed relations between assembly objects are presented in Section 3.2. 3.1 Definition of Assembly Objects In 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 an attribute of O, associated with each attribute is a type, t, and a value, v. M is a set of tuples, (m, tc 1 , tc 2 , %, tc n , tc). Each element of M is a function that uniquely identifies a method. The symbol m represents a method name; and methods define operations on objects. The symbol tc i (i Fig. 2. Feature-based, object-oriented hierarchical representation. 742 X. G. Ye et al. = 1, 2, %, n) specifies the argument type and tc specifies the returned value type. R is a set of relationships among O and other assembly objects. There are six types of basic relationships between 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 are attributes with a data type of string; checkFinterference and pocketFplate are member functions; and Part-of, SR and DOF are relationships. 3.2 Assembly Relationships There are six types of basic relationships between assembly objects, Part-of, SR, SC, DOF, Lts, and Fit. Part-of An assembly object belongs to its ancestor object. SR Spatial relations: explicitly specify the positions and orientations of assembly objects in an assembly. For a component part, its spatial relationship is derived from spatial constraints (SC). SC Spatial constraints: implicitly locate a component part with respect to the other parts. DOF Degrees of freedom: are allowable translational/ rotational directions of motion after assembly, with or without limits. Lts Motion limits: because of obstructions/interferences, the DOF may have unilateral or bilateral limits. Fit Size constraint: is applied to dimensions, in order to maintain a given class of fit. Table 1. Definition of an assembly object-ejector. Object Oid ejector-pinF1 Instance-of EjectorFpin Derived from ejector class A Purpose “ejecting moulding” Type string Material “nitrided steel” Type string CatalogFno “THX” Type string M CheckFinterference Check interference (coolFobj) between ejectors and cooling lines PocketFplate() Make a hole on plate to accommodate ejector pins R Part-of ejectorFsys SR align with EB plate DOF Tx, Ty Among all the elements of an assembly object, the relation- ships are most important for assembly design. The relationships between assembly objects will not only determine the position of objects in an assembly, but also maintain the associativities between assembly objects. In the following sub-sections, we will illustrate the relationships at the same assembly level with the help of examples. 3.2.1 Relationships Between Form Features Mould design, in essence, is a mental process; mould designers most of the time think on the level of real-world objects such as plates, screws, grooves, chamfers, and counter-bore holes. Therefore, it is necessary to build the geometric models of all product-independent parts from form features. The mould designer can easily change the size and shape of a part, because of the relations between form features maintained in the part representation. Figure 3(a) shows a plate with a counter-bore hole. This part is defined by two form features, i.e. a block and a counter-bore hole. The counter-bore hole (FF 2 ) is placed with reference to the block feature FF 1 , using their local coordinates F 2 and F 1 , respectively. Equations (2) (5) show the spatial relationships between the counter-bore hole (FF 2 ) and the block feature (FF 1 ). For form features, there is no spatial constraint between them, so the spatial relationships are specified directly by the designer. The detailed assembly relationships between two form features are defined as follows: SR(FF 2 ,FF 1 ): F 2i =- F 1i (2) F 2j =- F 1j (3) Fig. 3. Assembly relationships. Automated Assembly Modelling 743 F 2k = F 1k (4) r 2F = r 1F + b 22 *F 1j + A F1 *F 1i (5) DOF: ObjFhasF1FRDOF(FF 2 , F 2j ) The counter-bore feature can rotate about axis F 2j . LTs(FF 2 , FF 1 ): A F1 , b 11 - 0.5*b 21 (6) Fit (FF 2 , FF 1 ): b 22 = b 12 (7) Where F and r are the orientation and position vectors of fea- tures. F 1 = (F 1i , F 1j , F 1k ), F 2 = (F 2i , F 2j , F 2k ). b ij is the dimension of form features, Subscript i is feature number, j is dimension number. A F1 is the dimension between form features. Equations (2)(7) present the relationships between the form feature FF 1 and F