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黃河科技學院畢業(yè)設計(文獻翻譯) 第 16 頁 組合機床設計的模塊化建模方法 圖爾加·埃薩爾 美國密歇根大學研究生研究助理機械工程系 杰弗里L·斯坦 美國密歇根大學機械工程學系 盧卡斯·盧卡 塞浦路斯大學機械與制造工程講師部 摘要 在市場需求訊息萬變的情況下，為提升工業(yè)競爭力，稱為組合機床（RMTs）的新一代機床應運而生。為這些機床的高效設計，則須提出新的方法和工具。這是本文提出組合機床伺服軸模塊化建模方法的目的，而這也只是努力發(fā)展集成的組合機床設計和控制環(huán)境的一部分。該機床的組件被模塊化，這樣就可以將相應的組件基于機床拓撲學裝配起來得到任何特定的配置模型。組件模型使用內(nèi)置的圖形代碼以促進所需模塊庫的直接發(fā)展。這些機床模塊可用于評估，設計和機床伺服軸的控制。這種方法已有實踐證明，人們對其優(yōu)缺也有一定認識。結(jié)果表明，該方法是實現(xiàn)自動化和集成的機床設計環(huán)境很有希望的一步。人們對完成這個目標所要面對的挑戰(zhàn)也進行了探討。 引言 不斷增長的競爭迫使制造商更快速地響應需求的變化。因此，制造商必須面對產(chǎn)品市場周期短，過渡時期短，型號和量變化頻繁的情形，而且不能影響產(chǎn)品質(zhì)量和成本。 作為制造系統(tǒng)的核心，改進的機床在滿足上面提到的需求上把握著關(guān)鍵技術(shù)。傳統(tǒng)的機床在專用和柔性上的缺點今更勝昔：因其設計的重點在單一部件，使專用設備缺乏柔性機床所具備的靈活性和可擴展性。另一方面，柔性機床無法實現(xiàn)魯棒性，高的成本效益和專用設備所有的生產(chǎn)量水平[1]。 新一代機床在美國密歇根大學工程技術(shù)研究中心由Ann Arbor主持開發(fā)，為的是克服現(xiàn)有生產(chǎn)系統(tǒng)的不足部分為可重構(gòu)制造系統(tǒng)而開發(fā)。這些機床稱為組合機床（RMTs）[2]，它們結(jié)合了專業(yè)和靈活的優(yōu)點。它們圍繞一個零件族和他們的結(jié)構(gòu)設計，在硬件和軟件方面，可以快速的改變，經(jīng)濟有效地實現(xiàn)精確的功能和滿足設備需求 [3] 。含多種配置，提供所需柔性和可擴展性，RMTs本質(zhì)上導致了更復雜的機床設計問題。幫助促進RMTs設計的方法和工具將很大的促進可重構(gòu)制造系統(tǒng)的應用[4-6]。RMTs設計問題的一個重要方面是發(fā)展動態(tài)模型的設計，伺服軸的控制和賦值。使RMTs建模問題獨特的 是，即使僅有一臺機床，也和存在幾種不同的配置，且單獨的模式，必須開發(fā)。為所有可能的配置開發(fā)動態(tài)模型可能是一個繁瑣和費時的任務，即使利用了特別的方法。而且，沒有系統(tǒng)的方法建模將需要大量的專業(yè)知識，并容易出錯，從而降低了設計中使用模型的效率。 本文提出了一種方法，可以有助于減少RMTs建模的時間，出錯和麻煩。這一方法的核心思想是利用RMTs的模塊化結(jié)構(gòu)的優(yōu)勢，采取RMTs模塊化建模方法的建模概念。首先，RMTs的物理組件的模塊化建模方式是使用鍵合圖建模工具[7]。該鍵合圖模型被封裝在一個定義的連接端口的示意圖中。然后，原理組件模型按照給定的配置拓撲來組裝獲取配置模型。配置模型很容易與非動部件如插補器和控制器結(jié)合，這可用條形圖方便體現(xiàn)；但是這超出了本文的范圍。 背景 RMTs概念是由科倫和哥打[2]提出，從那時起，RMTs的設計就是一個活躍的研究領(lǐng)域。設計RMTs[4]的方法、工具以及評價結(jié)構(gòu)剛度[5]和提示錯誤[6]設計的替代工具已經(jīng)開發(fā)出來。然而，開發(fā)一個系統(tǒng)級建模方法的問題還沒有解決。傳統(tǒng)上，機床模型描繪伺服電機和驅(qū)動裝置為第一或第二階系統(tǒng)[8,9 ]。然而，陳和特盧斯季認為一旦采用了高速機床伺服驅(qū)動的結(jié)構(gòu)動力可能會影響系統(tǒng)性能 [10]。許多研究人員認定需要配合結(jié)構(gòu)動力學在高速機床上使用高階模型，以便能夠成功設計其控制系統(tǒng) [11-13]。這些論述清楚地表明，機床建模不是一項簡單的任務和考慮復雜模型時必須要細心，但他們沒有提供系統(tǒng)的建模方法，因此，仍然得應用特殊方法。 為有助于設計和控制機床伺服驅(qū)動，仍有人努力做自動仿真模型的研究。威爾遜和斯坦開發(fā)了一個叫建模助手的軟件程序能在一個給定的影響范圍內(nèi)自動創(chuàng)成機床驅(qū)動系統(tǒng)微型模型（FROI）[14]。該模型包括飛輪、一個扭轉(zhuǎn)軸、一滾珠絲杠、一滾珠直流電動機、一個扭轉(zhuǎn)連接鍵、帶驅(qū)動和齒輪副的組成部分，其復雜性自動增加，直至超過規(guī)定FROI的特征值。這項工作僅是一個概念模型推演法則的證明，不能適用于任何真正的機床系統(tǒng)。不過，這種方法可以用來確定發(fā)展系統(tǒng)模型時其復雜性是否適當。 戈蒂埃等已經(jīng)開發(fā)出一種名為SICOMAT的軟件包（仿真與控制的機床分析），這有助于建模，仿真，模態(tài)分析和控制器的一倍或兩倍減震或兩個機床軸耦合[15]。他們的模型由大量的模塊和彈簧描述機械系統(tǒng)的動態(tài)特性。這項工作使得機床的建模過程更加系統(tǒng)，因此對建模工程師來講是很有價值的工具，但它缺乏RMTs設計方式所要求的普遍性、模塊性和柔性。RMTs建模方法如圖1顯示了所設想的RMTs建模環(huán)境。這對于實現(xiàn)RMTs建模任務自動化是理想的，而特定的RMTs的配置模型自動從標準組件模塊庫組裝。這樣所有人工或自動產(chǎn)生的候選設計都可以快速模擬，而其模型可用于就它們的伺服軸動態(tài)性能和幫助設計方面評價候選方案，如圖一所示，模塊化組件模型庫是一個自動的RMT建模環(huán)境的重要組成部分。因此，擬議方式的第一步就是為了開發(fā)用于生成RMT配置組件的標準模型。本文將重點放在機械零件上，并討論了它們模塊化的建模方法。因為機械部件之間相互影響，促成它們的模塊化,更有趣的造型。只交流如插補器和控制器信號的模塊化建模的組件，提出了一種比較簡單的問題，這兒不再討論。為促進模塊化和使組件與環(huán)境之間的能量交流更容易，鍵合圖被作為了建模語言。鍵合圖提供電力的物理系統(tǒng)的圖形表示。此外，鍵合圖用統(tǒng)一的方式描述了不同的能源領(lǐng)域，這對RMTs建模是相應的優(yōu)勢，因為他們的伺服軸可能包括來自不同領(lǐng)域如機械、電氣或液壓的組件。鍵合圖只是用在這項工作中模型體現(xiàn)分級結(jié)構(gòu)中的一級。鍵合圖下一水平的數(shù)學方程式代表鍵合圖體現(xiàn)的物理現(xiàn)象，這種數(shù)學體現(xiàn)只是層次結(jié)構(gòu)中的最低水平。最高級別的鍵合圖都被封裝在一個示意圖中，這不僅表現(xiàn)緊湊，而且還顯示與環(huán)境融合的連接端口。圖2說明了這種層次模式體現(xiàn)。 這篇論文所有的模型顯示在原理層次，因為本文的目的不是討論他們的來歷，而是想一旦得到這些模型能夠做些什么。本文所用模型的詳細描述在[16]中都可以找到。為了能夠應付任何經(jīng)歷不同配置的機械部件的空間運動，使用了捕捉三維動態(tài)的模型。此外，最初的假設是，在機械領(lǐng)域范圍內(nèi)所有組成部分都可作為剛體充分體現(xiàn)。 圖3所示的有N連接端口的一般剛體是模塊庫中的一個主要模塊。對應于剛體上興趣點的端口，與環(huán)境發(fā)生物理上的交互。關(guān)系用于指示端口是原子端口，如主要部分能通過端口與所處環(huán)境交換能量。而現(xiàn)行的關(guān)系則指出了信號端口。只有信息通過這些端口傳輸。模型庫還包含三維連接模塊可用于描述構(gòu)件模型的相關(guān)運動。這些聯(lián)合模塊和端口也以標準開發(fā)，這樣他們可以連接到其他模型模塊。該庫提供了兩種方法來表達制約因素：（1）硬的彈簧和減震器可以用來實現(xiàn)更現(xiàn)實的限制，或近似理想的限制；（2）拉格朗日乘數(shù)可以引進來表達理想約束。對于聯(lián)合模塊的論述讀者也被稱為[16]。一旦模型庫由一些基本的模塊化剛體和模型組建，建模過程可以進行如下：RMT構(gòu)件被分解成單件，每個單件與庫中的模型相關(guān)。如果庫中的模型模塊都不能完全描述這個單件，那么必須開發(fā)一個新的相關(guān)模塊并添加到庫中。然后，模型根據(jù)構(gòu)件拓撲并使用必要的結(jié)合塊組裝。一旦獲得組件模型，它可以存儲在庫中備用。最后，組件模型按照給定的配置拓撲獲得該配置模型的組裝。這個過程可以用圖4的流程圖及以下部分的例子給予證明。 實例 下面兩個例子給提議的建模方法一個概述。第一個例子顯示了一個幻燈片建模和第二個例子采用該幻燈片模式發(fā)展為RMTs模式。這些例子的目的是提供一個關(guān)于組件的模塊化如何用在建模流程中的總體思路，而不是解釋每個（子）組件如何識別和建模的詳細信息。因此，該模型模塊的細節(jié)如它們的復雜程度都沒有討論。 滑動體建模是大多數(shù)機床工具基本組成部分，包括RMTs。不同的RMTs配置可以通過添加/刪除滑動體獲得或重新配置現(xiàn)有結(jié)構(gòu)中的滑動體。因此演示一個滑動體建模流程是有益的。參看圖5所示的滑動體。這是假設的構(gòu)成部分即如圖所示。本示例的目的，所有除電機外的子構(gòu)件可以像剛體與各連接點樣建模。電動機的動力可分為兩個用途：三維架構(gòu)剛體動力和驅(qū)動轉(zhuǎn)子和定子之間旋轉(zhuǎn)運動的機電動力。電動機獲取雙領(lǐng)域動力的模型也已經(jīng)開發(fā)出來，其示意圖由圖6給出。 弓形RMT的模型是由美國國家科學基金會工程研究中心可重構(gòu)制造系統(tǒng)在密歇根大學開發(fā)的，是世界上第一個完整規(guī)模的RMTs。這是一個3軸機床，其設計圍繞著具有五個不同表面族，這些表面傾角變化范圍從-15 °至45 °，一次增量15 °。它還具有如磨削、鉆削加工任意角度的柔性。弓形RMT的可重構(gòu)性來自于主軸單元，它可通過弓形模塊的彎曲導向槽移動從而在上面提到的5個角度得到配置，然后在任意一個位子上由機械擋塊固定。為了舉例假設基本模塊是完全相同的，并對機床沒有動力的影響。該工作臺、圓柱、主軸基本上都是滑動體，其模型都以上述滑動體模型為基礎。該拱被建模為剛體并帶有與每個機械擋塊連接的端口。最后，該拱式RMT模型按實際機器的拓撲結(jié)構(gòu)組裝。要注意這個數(shù)字顯示的模型只是配置之一。其他配置的模型可通過改變拱模型連接端口得到。 既然模型已組裝，運動方程可以自動從圖形模式得出，并執(zhí)行仿真。盡管數(shù)學模型準備好了，由于當前缺乏好的系統(tǒng)參數(shù)估計，我們不能在本文中提供任何仿真結(jié)果。一旦參數(shù)值可用仿真就很容易進行。 討論 本文中標準建模和分級建模概念被確定為RMTs建模方法的主要特點。RMTs的模塊化結(jié)構(gòu)使這種建模方法很有益處，因為這些模型包含了所有可重構(gòu)的重要特征[17]： 1．模塊化：（子）組件建模模塊化 2．可集成：該模塊可以通過其連接端口與其他模塊集成 3．定制：詳細程度包括了模型模塊都可為單個組件進行定制 4．可重構(gòu)性：模型可以很容易地從一個配置轉(zhuǎn)換到另一個 5．診斷性：可方便地進行模塊的模型驗證 本文介紹的方法可以將建模任務分為兩個步驟：（1）開發(fā)組件模型；（2）裝配配置模型。雖然第一步仍然需要大量的建模知識，第二步更系統(tǒng)，甚至將來要實現(xiàn)自動化。此外，兩個步驟各有側(cè)重：第一步的重點是組件內(nèi)動態(tài)，而第二個步驟重點是組件之間的動態(tài)。 相對于伺服軸建?，F(xiàn)有的方法，各個不同的RMTs配置都是一個潛在的新建模問題，本文提出的方法允許配置模式更快的發(fā)展。配置可以使用庫中的模型模塊快速組裝，假若如此，使用在給定配置的所有組件在模庫中都有對應的模塊。因此，一個完善的模型庫對這一方法的有效性是必不可少的。 對機床的機械部件三維多體建模方法推動了機械領(lǐng)域的模塊化。因此，機床滑動模型可用于任何配置，例如在一個滑臺移動有更多約束的環(huán)境下而不需要特別的滑臺模型。以多體的方法，開發(fā)通用組件模型可以沒有組件接口的先驗知識。然而三維多體方法的一個缺點是通用模型可能比某一實際配置的需求更復雜。例如，在給定的配置中一個組件可能只是被限制在一個平面運動中，在這種情況下，三維模型就過復雜了。該模型應該簡化，否則模型中不必要的復雜性降低了模型的計算效率。擬議的模塊化建模方法將從集成模型降階算法中受益。這將是今后工作的重點。 目前，該機構(gòu)被認為是剛性，這并不總是很接近。為了能夠研究結(jié)構(gòu)動力學的影響，柔性的機構(gòu)模式也應開發(fā)并添加到模庫內(nèi)。 最后，值得注意的是商業(yè)可用軟件包如ADAMS,、DADS、 EASY5、 Dymola 等也可用于RMTs建模的目的。 然而，采取統(tǒng)一強大的鍵合圖提供的功能 基礎方法，并簡化未來的模式使其更易于實施，那么鍵合圖就被選為建模語言。 概要和結(jié)論 模塊化建模方法被提議作為組合機床建模方法。這些部件在標準方式下建模，這么做的目的就是為了在組建給定的機床配置模型時只需將相應的模塊組裝就可以了。本文給出了兩個例子來說明這種方法，并且討論了這種方法的優(yōu)劣。 這項工作的結(jié)果表明，RMTs的模塊化建模問題可以使建模過程系統(tǒng)化，這樣對于在職工程師來講要獲得自動的設計建模環(huán)境有潛在的用處。但是如在討論中所強調(diào)的那樣，仍然有很多挑戰(zhàn)需要在自動化建模環(huán)境切實執(zhí)行之前落實。 致謝 這項工作是在歐洲經(jīng)濟共同體9529125授權(quán)下由國家科學基金會可重構(gòu)制造系統(tǒng)工程技術(shù)研究中心支持的。 A MODULAR MODELING APPROACH FOR THE DESIGN OF RECONFIGURABLE MACHINE TOOLS Tulga Ersal Graduate Student Research Assistant Department of Mechanical Engineering University of Michigan Jeffrey L. Stein Professor Department of Mechanical Engineering University of Michigan Loucas . Louca Lecturer Department of Mechanical and Manufacturing Engineering University of Cyprus ABSTRACT A new generation of machine tools called Reconfigurable Machine Tools (RMTs) is emerging as a means for industry to be more competitive in a market that experiences frequent changes in demand. New methodologies and tools are necessary for the efficient design of these machine tools. It is the purpose of this paper to present a modular approach for RMT servo axis modeling, which is part of a larger effort to develop an integrated RMT design and control environment. The components of the machine tool are modeled in a modular way, such that the model of any given configuration can be obtained by assembling the corresponding component models together based on the topology of the machine. The component models are built using the bond graph language that enables the straightforward development of the required modular library. These machine tool models can be used for the evaluation, design and control of the RMT servo axes. The approach is demonstrated through examples, and the benefits and drawbacks of this approach are discussed. The results show that the proposed approach is a promising step towards an automated and integrated RMT design environment, and the challenges in order to complete this goal are discussed. INTRODUCTION The ever-growing competition forces manufacturers to respond more quickly to changes in demand. As a result, manufacturers have to deal with short product life cycles, short ramp-up times and frequent changes in product mix and volumes, without compromising product quality and cost. Being the heart of a manufacturing system, improved machine tools hold the key in meeting the above mentioned requirements. The shortcomings of conventional machine tools, which can be classified as dedicated and flexible, are being felt more today than in the past: With their design focus being a single part, dedicated machines lack the flexibility and scalability that the flexible machines offer. On the other hand, flexible machines cannot achieve the robustness, the cost-effectiveness and the throughput levels of dedicated machines[1]. A new generation of machine tools is being developed in the Engineering Research Center for Reconfigurable Manufacturing Systems at the University of Michigan, Ann Arbor, as part of an effort to overcome the insufficiencies of current manufacturing systems. These machine tools are called ‘Reconfigurable Machine Tools (RMTs)’ [2], and they combine the advantages of their dedicated and flexible counterparts. They are designed around a part family and their structure, in terms of both hardware and software, can be changed quickly and cost-effectively to achieve the exact functionality and capacity desired [3]. Containing several configurations to provide the needed flexibility and scalability, RMTs intrinsically lead to more complex machine tool design problems. Methodologies and tools that would help facilitate the design of RMTs could highly benefit and encourage the employment of reconfigurable manufacturing systems [4-6]. One important aspect of the RMT design problem is developing dynamic models for the design, evaluation and control of servo axes. What makes the problem of modeling RMTs unique is that even though there is a single machine tool, there exist several configurations, which separate models have to be developed for. Developing dynamic models for all possible configurations could be a cumbersome and time-consuming task if ad hoc methods are utilized. Moreover, without a systematic methodology modeling would require a lot of expertise and would be prone to errors, which would degrade the efficiency of using models in the design. In this paper we present a methodology that could help make the RMT modeling task less time demanding, less error-prone and less challenging. The key idea of this methodology is to take advantage of the modular structure of the RMTs and adopt modular modeling concepts into the RMT modeling methodology. First, the physical components of an RMT are modeled in a modular way using the bond graph modeling tool [7]. The bond graph model is encapsulated in a schematic representation with defined connection ports. Then, the schematic component models are assembled by following the topology of a given configuration to obtain the model of the configuration. The configuration model can be easily integrated with the modules of non-energetic components such as interpolators and controllers, which can be conveniently represented with block diagrams; however this is beyond the scope of this paper. BACKGROUND The RMT concept was introduced by Koren and Kota [2], and since their introduction, the design of RMTs has been an active research area. Methodologies and tools for designing RMTs [4] as well as evaluating structural stiffnesses [5] and tool tip errors [6] of de sign alternatives have been developed. However, the problem of developing a system level modeling methodology for RTMs has not been addressed yet. Traditionally, machine tool models depict the machine tool as a group of servomotor and feed drive assemblies that aremodeled as first or second order systems [8,9]. Chen and Tlusty, however, showed that the structural dynamics of the feed drive could affect the system performance once high-speed machine tools are considered [10]. Many researchers identified the necessity to use higher order models for high-speed machine tools to cope with structural dynamics in order to be able to design the control system successfully [11-13].These publications clearly indicate that modeling a machine tool is not a trivial task and care must be taken when deciding on the complexity of the model, but they do not provide a systematic way of modeling and, therefore, remain application specific approaches. There have been research efforts to help the design and control of machine tool feed drives by automatically providing simulation models. Wilson and Stein developed a software program called Model-Building Assistant to automatically synthesize a minimum order model of the machine tool drive system for a given frequency range of interest (FROI) [14]. The complexity of the model, which includes a flywheel, a torsional shaft, a ballscrew , a ballnut, a DC motor, a torsional coupling, a belt-drive and a gear-pair as components, is automatically increased until the eigenvalues of the system fall beyond the specified FROI. This work was a proof of concept for a model deduction algorithm and can not be applied to any real machine tool system. However, such algorithm can be used to determine the appropriate model complexity after the development of the system model. Gautier et al. have developed a software package called SICOMAT (Simulation and Control analysis of Machine Tools) which helps with the modeling, simulation, modal analysis and controller tuning of one or two decoupled or two coupled machine tool axes [15]. Their models describe the dynamics of the mechanical system by a number of masses and springs. This work makes the modeling of a machine tool process more systematic, and is therefore a valuable tool to the modeling engineer; however, it lacks the generality, modularity and flexibility that the RMT design methodology demands. The RMT modeling methodology Figure1 shows the envisioned RMT modeling environment. It is desired to automate the task of RMT modeling, where the model of a given RMT configuration is automatically assembled from a library of modular component models. This way, all the candidate designs, which are generated either manually or automatically [4], can be modeled quickly and the models can be used to evaluate the candidates in terms of their servo axis dynamic performance and help with their design. As Figure1 also implies, the modular component model library is a key part for the automated RMT modeling environment. Therefore, the first step of the proposed methodology is to develop modular models for the components that are used to generate the RMT configurations. This paper puts the emphasis on mechanical parts and discusses their modeling in a modular way, because the energy interaction between the mechanical components makes their modular modeling more intriguing. Modular modeling of components that only exchange signals, e.g. interpolators and controllers, presents a relatively simpler problem and are not discussed here. To promote modularity and to be able to deal with the energy interactions between the components and their environment rather easily, bond graphs are utilized as the modeling language. Bond graphs provide a power-based graphical representation of a physical system. Moreover, bond graphs describe different energy domains in a unified way, which is a relevant advantage for RMT modeling, since their servo axes may include components from different energy domains, such as mechanical, electrical or hydraulic. Bond graphs are only one level in the hierarchy of model representations used in this work. Underneath the bond graph level the mathematical equations represent the physical phenomena captured by the bond graph and this mathematical representation is the lowest level in the hierarchy. In the highest level bond graphs are encapsulated in a schematic representation, which not only allows for a compact representation, but also shows the connection ports where the model can interact with its environment. Figure 2 illustrates this hierarchy of model representations. In this paper all the models are shown in the schematic level, because the goal of this paper is not to discuss their derivation, but rather to show what can be done once those models are obtained. A detailed description of the models used in this paper can be found in [16].In order to be able to cope with any spatial motion that the mechanical components may go through in different configurations, models that capture the three-dimensional dynamics are used. Moreover, the initial assumption is made that in the mechanical domain all components can be adequately represented as rigid bodies. Figure 3 shows the schematic representation of a generic rigid body with N connection ports, which is one of the main model modules in the library. The ports correspond to points of interest on the rigid body, where the physical interactions with the environment occur. Bonds (lines with half arrows) are used to indicate that a port is a power port, i.e. the body can exchange energy with its environment through those ports, whereas active bonds (lines with full arrows) indicate signal ports, i.e. only information is transferred through these ports. The model library also contains three-dimensional joint models that can be used to describe the relative motions between the component models. These joint models are also developed in a modular way with ports, where they can be connected to other model modules. The library offers two ways to express the constraints: (1) stiff springs and dampers can be used to implement more realistic constraints or to approximate ideal constraints;(2) Lagrange multipliers can be introduced to express the constraints ideally. For a discussion of joint models the reader is also referred to [16].Once the model library is populated with some basic modular rigid body and joint models, the modeling procedure can be carried out as follows: The RMT components are broken down into subcomponents and each subcomponent is associated with a model in the library. If none of the model modules in the library can describe the subcomponent adequately, a new model has to be developed for that subcomponent and added to the library. Then, the models are assembled by following the topology of the components and using the necessary joint models. Once a component model is obtained, it can be stored in the library for reuse. Finally, the component models are assembled by following the topology of a given configuration to obtain the model of that configuration. The process is illustrated in Figure 4 as a flowchart and demonstrated in the following section through examples. EXAMPLES The following two examples give an overview of the proposed modeling methodology. The first example shows the modeling of a slide and the second example employs that slide model to develop a model for a RMT. The purpose of these examples is to give a general idea about how the modularity of the components can be exploited in the modeling procedure, rather than to explain the details of how each (sub)component can be identified and modeled. Therefore, the details of the model modules, such as their level of complexity, are not discussed. Modeling a Slide A slide is a basic component of most machine tools, including RMTs. Different RMT configurations can beobtained by adding/removing slides to/from the configuration or by rearranging the existing slides in the configuration. Therefore, it is useful to demonstrate the modeling procedure of a slide. Consider the slide shown in Figure 5. It is assumed that the components are identified as shown in the figure. For the purposes of this example, all the subcomponents except the motor can be modeled as rigid bodies with various number of connection points. The motor dynamics can be broken down into two domains: the three-dimensional rigid body dynamics of the housing and the electromechanical dynamics that drive the relative rotational motion between the rotor and the stator. A model has been developed for the motor that captures the dynamics in both domains and its schematic representation is given in Figure 6. Modeling the Arch-type RMT, which was developed by the NSF Engineering Research Center for Reconfigurable Manufacturing Systems at the University of Michigan, is the world’s first full scale RMT. It is a three-axis machine tool that is designed around a part family with five different surface inclinations ranging from -15° to 45° at 15° increments and has the flexibility of doing machining operations such as milling and drilling at any of those angles. The reconfigurability of the Arch-type RMT comes from the spindle unit, which can be configured at the five angles mentioned above by moving it along the curved guide way of the arch module and fixing it at any of the five locations on the arch module that are defined by mechanical stops. For the purposes of this example the base module is assumed to be identical to the ground and it has no effect on the dynamics of the machine tool. The worktable, the