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南京理工大學(xué)泰州科技學(xué)院 學(xué)生畢業(yè)設(shè)計（論文）中期檢查表 學(xué)生姓名 錢瑞 學(xué) 號 0501510131 指導(dǎo)教師 武培軍 吳晟 選題情況 課題名稱 集裝箱波紋板焊接機器人機構(gòu)運動學(xué)分析 及車體結(jié)構(gòu)設(shè)計 難易程度 偏難 適中 √ 偏易 工作量 較大 合理 √ 較小 符合規(guī)范化的要求 任務(wù)書 有 √ 無 開題報告 有 √ 無 外文翻譯質(zhì)量 優(yōu) 良 √ 中 差 學(xué)習(xí)態(tài)度、出勤情況 好 √ 一般 差 工作進度 快 按計劃進行 √ 慢 中期工作匯報及解答問題情況 優(yōu) 良 √ 中 差 中期成績評定：良 所在專業(yè)意見： 學(xué)習(xí)較主動、積極，態(tài)度認真，階段成果較明顯。 負責(zé)人： 年 月 日 南京理工大學(xué)泰州科技學(xué)院 畢業(yè)設(shè)計說明書(論文) 作 者: 錢瑞 學(xué) 號： 0501510131 系　部: 機械工程系 專 業(yè): 機械工程及自動化 題 目: 集裝箱波紋板焊接機器人機構(gòu) 運動學(xué)分析及車體結(jié)構(gòu)設(shè)計 助教 吳晟 指導(dǎo)者： 評閱者： 2009 年 6 月 畢業(yè)設(shè)計說明書（論文）中文摘要 隨著工業(yè)水平的發(fā)展,重要的大型焊接結(jié)構(gòu)件的應(yīng)用越來越多,其中大量的焊接工作必須在現(xiàn)場作業(yè),如集裝箱波紋板焊接機器人、大型艦船艙體、甲板的焊接、大型球罐(儲罐)的焊接等。而這些焊接場合下，焊接機器人要適應(yīng)焊縫的變化，才能做到提高焊接自動化的水平。無疑,將機器人技術(shù)和焊縫跟蹤技術(shù)結(jié)合將有效地解決大型結(jié)構(gòu)件野外作業(yè)的自動化焊接難題。因此機器人的設(shè)計對于解決這一難題至關(guān)重要。 本課題主要完成機器人運動學(xué)的逆解、車體的總體設(shè)計、電機的選擇等方面。主要從機器人運動學(xué)逆解的角度完成一個周期內(nèi)的運動學(xué)逆解，求出三個關(guān)節(jié)應(yīng)按照什么運動規(guī)律進行運動，還有三個關(guān)節(jié)的運動之間的函數(shù)關(guān)系，進而完成對整個機器人的總體設(shè)。通過對小車的受力分析完成對車輪、車體的設(shè)計。根據(jù)實際操作中遇到的問題對完成對電機的選擇。最后對所選的齒輪進行校核，使之能完成具體的操作要求。 關(guān)鍵詞 機器人技術(shù) 機構(gòu)設(shè)計 運動學(xué)逆解 強度校核 畢業(yè)設(shè)計說明書（論文）外文摘要 Title Robot developt Abstract With the development of industrial level, it is important to large-scale structure of the application of welding more and more, including a large number of welding operations must be at the scene, such as robot welding corrugated containers, large ship cabin, the deck of the welding, a large spherical tank (tank), such as welding. These welding occasion, the welding robot to adapt to changes in weld, welding can be done to improve the level of automation. There is no doubt that technology and robot seam tracking technology to effectively solve large-scale structure of the automation field welding problems. Therefore the design of the robot is essential for the solution to this problem. The main subject of the completion of the robot inverse kinematics solution, the body design, the choice of motor and so on. The main robot inverse kinematics from the perspective of a cycle through the end of the known actuator position posture against the solution of the coordinates of the joints, and then completed the whole set up of the robot. Force analysis of the completed car wheels, car body design. According to the actual problems encountered in the operation of the completion of the motor choice. Finally, the gear selected for verification, so that it can complete the specific operating requirements. Keywords Robot technology Organization is designed The kinematics goes against solution he intensity is proofreaded 南京理工大學(xué)泰州科技學(xué)院 畢業(yè)設(shè)計（論文）任務(wù)書 系　　　部： 機械工程系 專 業(yè)： 機械工程及自動化 學(xué) 生 姓 名： 錢瑞 學(xué) 號： 0501510131 設(shè)計(論文)題目： 集裝箱波紋板焊接機器人機構(gòu) 運動學(xué)分析及車體結(jié)構(gòu)設(shè)計 起 迄 日 期: 2009年 3 月09日 ~ 6月14日 設(shè)計(論文)地點: 南京理工大學(xué)泰州科技學(xué)院 指 導(dǎo) 教 師: 武培軍 吳 晟 專業(yè)負責(zé)人: 龔光容 發(fā)任務(wù)書日期: 2009年 2 月 26 日 畢 業(yè) 設(shè) 計（論 文）任 務(wù) 書 1．本畢業(yè)設(shè)計（論文）課題應(yīng)達到的目的： 針對集裝箱波紋板焊接自動化水平低的現(xiàn)狀，目前用于焊接集裝箱側(cè)板與頂側(cè)梁、底側(cè)梁的自動焊專機，由于在焊接過程中，焊槍不能隨波形的變化調(diào)整與焊槍速度的夾角（焊接工藝參數(shù)也未有變化），直接導(dǎo)致焊縫成形不能保持一致，進而影響焊縫的質(zhì)量。 該課題能有效的解決焊接過程中焊槍速度與波形夾角的問題，使焊接速度始終與波形垂直，進而保證焊接的穩(wěn)定性，提高焊接成形的一致性，提高焊接質(zhì)量。 2．本畢業(yè)設(shè)計（論文）課題任務(wù)的內(nèi)容和要求（包括原始數(shù)據(jù)、技術(shù)要求、工作要求等）： 本課題是集裝箱波紋板焊接機器人機構(gòu)運動學(xué)分析及車體機構(gòu)設(shè)計，通過十字滑塊選用，進而組成的焊接機器人能夠解決波內(nèi)斜邊段焊縫外觀成形與直線段焊縫不一致的問題。研究內(nèi)容如下： 1、在廣泛調(diào)研的基礎(chǔ)上，熟悉機器人的應(yīng)用的現(xiàn)場環(huán)境，明確設(shè)計目標； 2、設(shè)計出該焊接機器人的機構(gòu)方案，并對其進行運動學(xué)逆界，證明所選方案可行； 3、設(shè)計出小車車體結(jié)構(gòu)，并在圖紙上繪制出機器人的裝配圖。 應(yīng)達到的技術(shù)要求如下： 1、所求焊接過渡段中的過渡運動能較好的銜接直線段與波內(nèi)斜邊段的運動； 2、三自由度焊接機器人三個運動關(guān)節(jié)可按照一定的運動規(guī)律協(xié)調(diào)動作。 畢 業(yè) 設(shè) 計（論 文）任 務(wù) 書 3．對本畢業(yè)設(shè)計（論文）課題成果的要求〔包括畢業(yè)設(shè)計論文、圖表、實物樣品等〕： 1、相關(guān)資料的英文翻譯與文獻綜述； 2、焊接機器人的機構(gòu)方案； 3、小車車體結(jié)構(gòu)設(shè)計及相關(guān)零件圖、裝配圖； 4、畢業(yè)設(shè)計論文。 畢業(yè)設(shè)計成果以設(shè)計圖樣和說明書形式提交。要求圖樣規(guī)范，符合國家標準；說明書層次分明、論據(jù)可靠、計算正確、圖標規(guī)范、語句通順。 4．主要參考文獻： [1]　鄭相鋒，胡小建．弧焊機器人焊接區(qū)視覺信息傳感與控制技術(shù)[J]．電焊機，2005． [2]　孔宇，戴明，吳林．機器人結(jié)構(gòu)光視覺三點焊縫定位技術(shù)[J]．焊接學(xué)報，1997． [3]　王軍波等．基于CCD傳感器的球罐焊接機器人焊縫跟蹤[J]．焊接學(xué)報，2001． [4]　徐培全等．基于機器人焊接的視覺傳感系統(tǒng)綜述[J]．焊接，2005． [5]　劉蘇宜，王國榮，鐘繼光．視覺系統(tǒng)在機器人焊接中的應(yīng)用與展望[J]．機械科學(xué)與技術(shù)，2005． [6]　張柯等．移動焊接機器人的研究現(xiàn)狀及發(fā)展趨勢[J]．焊接，2004． [7]　王其?。『高^程質(zhì)量實時傳感與控制[M]．北京：機械工業(yè)出版社，2000． 　 畢 業(yè) 設(shè) 計（論 文）任 務(wù) 書 5．本畢業(yè)設(shè)計（論文）課題工作進度計劃： 起 迄 日 期 工 作 內(nèi) 容 2009年 3 月 9 日 ~ 3月 23 日 3 月 24 日 ~ 4月 6 日 4 月 7 日 ~ 5月 11日 5 月 12 日 ~ 5月 31 日 6 月 1 日 ~ 6月 9 日 6月 10日 ~ 6月 14 日 熟悉課題，準備相關(guān)資料，完成資料翻譯 完成文獻綜述，撰寫開題報告，熟悉AutoCAD、Pro/Engineer等繪圖軟件 掌握焊接機器人的結(jié)構(gòu)設(shè)計原理，完成三自由度焊接機器人的運動學(xué)逆解 完成車體結(jié)構(gòu)設(shè)計，畫出相關(guān)的零件圖和裝配圖 撰寫并打印設(shè)計說明書，整理相關(guān)資料 準備論文答辯 所在專業(yè)審查意見： 負責(zé)人： 2008年 月 日 系部意見： 系部主任： 2008年 月 日 南京理工大學(xué)泰州科技學(xué)院 畢業(yè)設(shè)計(論文)前期工作材料 學(xué)生姓名: 錢瑞 學(xué) 號： 0501510131 系　部: 機械工程系 專 業(yè): 機械工程及自動化 設(shè)計(論文)題目: 集裝箱波紋板焊接機器人機構(gòu) 運動學(xué)分析及車體結(jié)構(gòu)設(shè)計 指導(dǎo)教師: 武培軍高工 吳晟助教 材 料 目 錄 序號 名 稱 數(shù)量 備 注 1 畢業(yè)設(shè)計(論文)選題、審題表 1 2 畢業(yè)設(shè)計(論文)任務(wù)書 1 3 畢業(yè)設(shè)計(論文)開題報告〔含文獻綜述〕 1 4 畢業(yè)設(shè)計(論文)外文資料翻譯〔含原文〕 1 5 畢業(yè)設(shè)計（論文）中期檢查表 1 2009年5月 南京理工大學(xué)泰州科技學(xué)院 畢業(yè)設(shè)計(論文)外文資料翻譯 系　　部： 機械工程系 專 業(yè)： 機械工程及自動化 姓 名： 錢 瑞 學(xué) 號： 0501510131 (用外文寫) 外文出處：The Internation Journal of Advanced Manufacturing Technology 附 件： 1.外文資料翻譯譯文；2.外文原文。 指導(dǎo)教師評語： 簽名： 年 月 日 注：請將該封面與附件裝訂成冊。 附件1：外文資料翻譯譯文 應(yīng)用坐標測量機的機器人運動學(xué)姿態(tài)的標定 這篇文章報到的是用于機器人運動學(xué)標定中能獲得全部姿態(tài)的操作裝置——坐標測量機（CMM）。運動學(xué)模型由于操作器得到發(fā)展, 它們關(guān)系到基坐標和工件。 工件姿態(tài)是從實驗測量中引出的討論, 同樣地是識別方法學(xué)。允許定義觀察策略的完全模擬實驗已經(jīng)實現(xiàn)。實驗工作的目的是描寫參數(shù)辨認和精確確認。用推論原則的那方法能得到在重復(fù)時近連續(xù)地校準機器人。 關(guān)鍵字：機器人標定 坐標測量 參數(shù)辨認 模擬學(xué)習(xí) 精確增進 1. 前言 機器手有合理的重復(fù)精度 (0.3毫米)而知名, 但仍有不好的精確性(10.0 毫米)。為了實現(xiàn)機器手精確性，機器人可能要校準也是好理解 。 在標定過程中， 幾個連續(xù)的步驟能夠精確地識別機器人運動學(xué)參數(shù)，提高精確性。這些步驟為如下描述： 1 操作器的運動學(xué)模型和標定過程本身是發(fā)展，和通常有標準運動學(xué)模型的工具實現(xiàn)的。作為結(jié)果的模型是定義基于廠商的運動學(xué)參數(shù)設(shè)置錯誤量, 和識別未知的,實際的參數(shù)設(shè)置。 2 機器人姿態(tài)的實驗測量法(部分的或完成) 是拿走為了獲得從聯(lián)系到實際機器人的參數(shù)設(shè)置數(shù)據(jù)。 3 實際的運動學(xué)參數(shù)識別是系統(tǒng)地改變參數(shù)設(shè)置和減少在模型階段錯誤量的定義。一個接近完成辨認由分析不同中間姿態(tài)變量P和運動學(xué)參數(shù)K的微分關(guān)系決定： 于是等價轉(zhuǎn)化得： 兩者擇一, 問題可以看成為多維的優(yōu)化問題，這是為了減少一些定義的錯誤功能到零點，運動學(xué)參數(shù)設(shè)置被改變。這是標準優(yōu)化問題和可能解決用的眾所周知的 方法。 4 最后一步是機械手控制中的機器人運動學(xué)識別和在學(xué)習(xí)之下的硬件系統(tǒng)的詳細資料。 包含實驗數(shù)據(jù)的這張紙用于標度過程。 可獲得的幾個方法是可用于完成這任務(wù), 雖然他們相當(dāng)復(fù)雜，獲得數(shù)據(jù)需要大量的成本和時間。這樣的技術(shù)包括使用可視化的和自動化機械 ，伺服控制激光干涉計，有關(guān)聲音的傳感器和視覺傳感器 。理想測量系統(tǒng)將獲得操作器的全部姿態(tài)(位置和方向)，因為這將合并機械臂各個位置的全部信息。上面提到的所有方法僅僅用于唯一部分的姿態(tài), 需要更多的數(shù)據(jù)是為了標度過程到進行。 2．理論 文章中的理論描述，為了操作器空間放置的各自的位置，全部姿態(tài)是可測量的，雖然進行幾個中間測量，是為了獲得姿態(tài)。測量姿態(tài)使用裝置是坐標測量機(CMM),它是三軸的，棱鏡測量系統(tǒng)達到0.01毫米的精確。機器人操作器是能校準的，PUMA 560，放置接近于CMM，特殊的操作裝置能到達邊緣。圖1顯示了系統(tǒng)不同部分安排。在這部分運動學(xué)模型將是發(fā)展, 解釋姿態(tài)估算法，和參數(shù)辨認方法。 2.1 運動學(xué)的參數(shù) 在這部分，操作器的基本運動學(xué)結(jié)構(gòu)將被規(guī)定，它關(guān)系到完全坐標系統(tǒng)的討論, 和終點模型。從這些模型，用于可能的技術(shù)的運動學(xué)參數(shù)的識別將被規(guī)定，和描述決定這些參數(shù)的方法。 那些基礎(chǔ)的模型工具用于描寫不同的物體和工件操作器位置空間的關(guān)系的方法是Denavit-Hartenberg方法，在Hayati 有調(diào)整計劃，停泊處 和當(dāng)二連續(xù)的接縫軸是名義上地平行的用于說明不相稱模型 。如圖2 這中方法存在于物體或相互聯(lián)系的操作桿結(jié)構(gòu)中，和運動學(xué)中需要從一個坐標到另一個坐標這種同類變化是被定義的。這種變化是相同形式的 上面的關(guān)系可以解釋通過四個基本變化操作實現(xiàn)坐標系n-1到結(jié)構(gòu)坐標系n的變化。只有需要找到與前一個的關(guān)系的四個變化是必需的，在那個時候連續(xù)的軸是不平行的，定義為零點。 當(dāng)應(yīng)用于一個結(jié)構(gòu)到下一個結(jié)構(gòu)的等價變化坐標系與更改Denavit-Hartenberg系相一致時，它們將被書寫成矩陣元素實現(xiàn)運動學(xué)參數(shù)功能的矩陣形狀。這些參數(shù)是變化的簡單變量：關(guān)節(jié)角，連桿偏置, 連桿長度，扭角，矩陣通常表示如下： 對于多連接的, 例如機械操作臂,各自連續(xù)的鏈環(huán)和兩者瞬間的位置描寫在前一個矩陣變化中。這種變化從底部鏈環(huán)開始到第n鏈環(huán)因此關(guān)系如下： 圖3表示出PUMA機器人在Denavit-Hartenberg系中每一連桿，完全坐標系和工具結(jié)構(gòu)。變化從世界坐標系到機器人底部結(jié)構(gòu)需要仔細考慮過，因為潛在的參數(shù)取決于被選擇的改變類型?？紤]到圖4,世界坐標，在D-H系中定義的從世界坐標到機器人基坐標，坐標是PUMA機器人定義的基坐標和機器人第二個D-H結(jié)構(gòu)中坐標。我們感興趣的是從世界坐標到必需的最小的參數(shù)數(shù)量。實現(xiàn)這種變化有兩種路徑：路徑1，從到D-H變化包括四個參數(shù)，接著從到的變化將牽連二個參數(shù)和的變化 圖3 圖4 最后，另外從到的D-H變化中有四個參數(shù)其中和兩個參數(shù)是關(guān)于軸Z0因此不能獨立地識別， 和是沿著軸Z0因此也不能是獨立地識別。因此，用這路徑它需要從世界坐標到PUMA機器人的第一個坐標有八個獨立的運動學(xué)參數(shù)。路徑2，同樣地二中擇一，從世界坐標到底部結(jié)構(gòu)坐標的變化可以是直接定義。因此坐標變換需要六個參數(shù)，如Euler形式： 下面是從到D－H變化中的四個參數(shù)，但與相關(guān)聯(lián)，與相關(guān)聯(lián)，減少成兩個參數(shù)。很顯然這種路徑和路徑1一樣需要八個參數(shù)，但是設(shè)置不同。 上面的方法可能使用于從世界坐標系到PUMA機器人的第二結(jié)構(gòu)的移動中。在這工作中，選擇路徑2。工具改變引起需要六個特殊參數(shù)的改變的Euler形式： 用于運動學(xué)模型的參數(shù)總數(shù)變成30，他們定義于表1 2.2 辨認方法學(xué) 運動學(xué)的參數(shù)辨認將是進行多維的消去過程, 因此避免了雅可比系統(tǒng)的標定，過程如下： 1. 首先假設(shè)運動學(xué)的參數(shù), 例如標準設(shè)置。 2. 為選擇任意關(guān)節(jié)角的設(shè)置。 3. 計算PUMA機器人末端操作器。 4. 測量PUMA機器人末端操作器的位姿如關(guān)節(jié)角，通常標準的和預(yù)言的位姿將是不同的。 5. 為了最好使預(yù)言位姿達到標準的位姿，在整齊的方式更改運動學(xué)的參數(shù)。 這個過程應(yīng)用于不是單一的關(guān)節(jié)角設(shè)置而是一定數(shù)量的關(guān)節(jié)角，與物理測量數(shù)量等同的全部關(guān)節(jié)角設(shè)置是需要，必須滿足 在這兒： Kp是識別的運動學(xué)參數(shù)的數(shù)量 N是測量位姿的數(shù) Dr是測量過程中自由度的數(shù)量 文章中，給定了自由度的數(shù)量，贈值為 因此全部位姿是測量的。在實踐中，更多的測量應(yīng)該是在實驗測量法去掉補償結(jié)果。優(yōu)化程序使用命名為ZXSSO，和標準庫功能的IMSL。 2.3 位姿測量法 顯然它是從上面的方法確定PUMA機器人全部位姿是必需的為了實現(xiàn)標定。這種方法現(xiàn)在將詳細地描寫。如圖5所示，末端操作器由五個確定的工具組成。 考慮到借助于工具坐標和世界坐標中間各個坐標的形式，如圖6 這些坐標的關(guān)系如下： 是關(guān)于世界坐標結(jié)構(gòu)的第i個球的4x1列向量坐標, Pi是關(guān)于工具坐標結(jié)構(gòu)第i個球的4x1坐標的列向量, T是從世界坐標結(jié)構(gòu)到工具坐標結(jié)構(gòu)變化的4x4矩陣。 設(shè)定Pi，測量出，然后算出T，使用于在標定過程的位姿的測量。它是不會很簡單，但是不可能由等式(11)反求出T。上面的過程由四個球A, B, C和D來實現(xiàn)，如下： 或為 由于P`, T和P全部相符合，反解求的位姿矩陣 在實踐中當(dāng)PUMA機器人放置在確定的位置上，對于CMM由四個球決定Pi是困難的。準確的測量三個球，第四球根據(jù)十字相乘可以獲得 考慮到?jīng)Q定的球中心坐標的是基于球表面點的測量,沒有分析可獲到的程序。 另外，數(shù)字優(yōu)化的使用是為了求懲罰函數(shù)的最小解 這里是確定球中心，是第個球表面點的坐標且是球的半徑。在測試過程中，發(fā)現(xiàn)只測量四個表面上的點來確定中心點是非常有效的。 附件2：外文原文（復(fù)印件） Full-Pose Calibration of a Robot Manipulator Using a Coordinate- Measuring Machine The work reported in this article addresses the kinematic calibration of a robot manipulator using a coordinate measuring machine (CMM) which is able to obtain the full pose of the end-effector. A kinematic model is developed for the manipulator, its relationship to the world coordinate frame and the tool. The derivation of the tool pose from experimental measurements is discussed, as is the identification methodology. A complete simulation of the experiment is performed, allowing the observation strategy to be defined. The experimental work is described together with the parameter identification and accuracy verification. The principal conclusion is that the method is able to calibrate the robot successfully, with a resulting accuracy approaching that of its repeatability. Keywords: Robot calibration; Coordinate measurement; Parameter identification; Simulation study; Accuracy enhancement 1. Introduction It is well known that robot manipulators typically have reasonable repeatability (0.3 ram), yet exhibit poor accuracy (10.0 mm). The process by which robots may be calibrated in order to achieve accuracies approaching that of the manipulator is also well understood . In the calibration process, several sequential steps enable the precise kinematic parameters of the manipulator to be identified, leading to improved accuracy. These steps may be described as follows: 1. A kinematic model of the manipulator and the calibration process itself is developed and is usually accomplished with standard kinematic modelling tools. The resulting model is used to define an error quantity based on a nominal (manufacturer's) kinematic parameter set, and an unknown, actual parameter set which is to be identified. 2. Experimental measurements of the robot pose (partial or complete) are taken in order to obtain data relating to the actual parameter set for the robot. 3.The actual kinematic parameters are identified by systematically changing the nominal parameter set so as to reduce the error quantity defined in the modelling phase. One approach to achieving this identification is determining the analytical differential relationship between the pose variables P and the kinematic parameters K in the form of a Jacobian, and then inverting the equation to calculate the deviation of the kinematic parameters from their nominal values Alternatively, the problem can be viewed as a multidimensional optimisation task, in which the kinematic parameter set is changed in order to reduce some defined error function to zero. This is a standard optimisation problem and may be solved using well-known methods. 4. The final step involves the incorporation of the identified kinematic parameters in the controller of the robot arm, the details of which are rather specific to the hardware of the system under study. This paper addresses the issue of gathering the experimental data used in the calibration process. Several methods are available to perform this task, although they vary in complexity, cost and the time taken to acquire the data. Examples of such techniques include the use of visual and automatic theodolites, servocontrolled laser interferometers , acoustic sensors and vidual sensors . An ideal measuring system would acquire the full pose of the manipulator (position and orientation), because this would incorporate the maximum information for each position of the arm. All of the methods mentioned above use only the partial pose, requiring more data to be taken for the calibration process to proceed. 2. Theory In the method described in this paper, for each position in which the manipulator is placed, the full pose is measured, although several intermediate measurements have to be taken in order to arrive at the pose. The device used for the pose measurement is a coordinate-measuring machine (CMM), which is a three-axis, prismatic measuring system with a quoted accuracy of 0.01 ram. The robot manipulator to be calibrated, a PUMA 560, is placed close to the CMM, and a special end-effector is attached to the flange. Fig. 1 shows the arrangement of the various parts of the system. In this section the kinematic model will be developed, the pose estimation algorithms explained, and the parameter identification methodology outlined. 2.1 Kinematic Parameters In this section, the basic kinematic structure of the manipulator will be specified, its relation to a user-defined world coordinate system discussed, and the end-point toil modelled. From these models, the kinematic parameters which may be identified using the proposed technique will be specified, and a method for determining those parameters described. The fundamental modelling tool used to describe the spatial relationship between the various objects and locations in the manipulator workspace is the Denavit-Hartenberg method , with modifications proposed by Hayati, Mooring and Wu to account for disproportional models when two consecutive joint axes are nominally parallel. As shown in Fig. 2, this method places a coordinate frame on each object or manipulator link of interest, and the kinematics are defined by the homogeneous transformation required to change one coordinate frame into the next. This transformation takes the familiar form The above equation may be interpreted as a means to transform frame n-1 into frame n by means of four out of the five operations indicated. It is known that only four transformations are needed to locate a coordinate frame with respect to the previous one. When consecutive axes are not parallel, the value of/3. is defined to be zero, while for the case when consecutive axes are parallel, d. is the variable chosen to be zero. When coordinate frames are placed in conformance with the modified Denavit-Hartenberg method, the transformations given in the above equation will apply to all transforms of one frame into the next, and these may be written in a generic matrix form, where the elements of the matrix are functions of the kinematic parameters. These parameters are simply the variables of the transformations: the joint angle 0., the common normal offset d., the link length a., the angle of twist a., and the angle /3.. The matrix form is usually expressed as follows: For a serial linkage, such as a robot manipulator, a coordinate frame is attached to each consecutive link so that both the instantaneous position together with the invariant geometry are described by the previous matrix transformation. 'The transformation from the base link to the nth link will therefore be given by Fig. 3 shows the PUMA manipulator with the Denavit-Hartenberg frames attached to each link, together with world coordinate frame and a tool frame. The transformation from the world frame to the base frame of the manipulator needs to be considered carefully, since there are potential parameter dependencies if certain types of transforms are chosen. Consider Fig. 4, which shows the world frame xw, y,, z,, the frame Xo, Yo, z0 which is defined by a DH transform from the world frame to the first joint axis of the manipulator, frame Xb, Yb, Zb, which is the PUMA manufacturer's defined base frame, and frame xl, Yl, zl which is the second DH frame of the manipulator. We are interested in determining the minimum number of parameters required to move from the world frame to the frame x~, Yl, z~. There are two transformation paths that will accomplish this goal: Path 1: A DH transform from x,, y,, z,, to x0, Yo, zo involving four parameters, followed by another transform from xo, Yo, z0 to Xb, Yb, Zb which will involve only two parameters ~b' and d' in the transform Finally, another DH transform from xb, Yb, Zb to Xt, y~, Z~ which involves four parameters except that A01 and 4~' are both about the axis zo and cannot therefore be identified independently, and Adl and d' are both along the axis zo and also cannot be identified independently. It requires, therefore, only eight independent kinematic parameters to go from the world frame to the first frame of the PUMA using this path. Path 2: As an alternative, a transform may be defined directly from the world frame to the base frame Xb, Yb, Zb. Since this is a frame-to-frame transform it requires six parameters, such as the Euler form: The following DH transform from xb, Yb, zb tO Xl, Yl, zl would involve four parameters, but A0~ may be resolved into 4~,, 0b, ~, and Ad~ resolved into Pxb, Pyb, Pzb, reducing the parameter count to two. It is seen that this path also requires eight parameters as in path i, but a different set. Either of the above methods may be used to move from the world frame to the second frame of the PUMA. In this work, the second path is chosen. The tool transform is an Euler transform which requires the specification of six parameters: The total number of parameters used in the kinematic model becomes 30, and their nominal values are defined in Table 1. 2.2 Identification Methodology The kinematic parameter identification will be performed as a multidimensional minimisation process, since this avoids the calculation of the system Jacobian. The process is as follows: 1. Begin with a guess set of kinematic parameters, such as the nominal set. 2. Select an arbitrary set of joint angles for the PUMA. 3. Calculate the pose of the PUMA end-effector. 4. Measure the actual pose of the PUMA end-effector for the same set of joint angles. In general, the measured and predicted pose will be different. 5. Modify the kinematic parameters in an orderly manner in order to best fit (in a least-squares sense) the measured pose to the predicted pose. The process is applied not to a single set of joint angles but to a number of joint angles. The total number of joint angle sets required, which also equals the number of physical measurement made, must satisfy Kp is the number of kinematic parameters to be identified N is the number of measurements (poses) taken Dr represents the number of degrees of freedom present in each measurement. In the system described in this paper, the number of degrees of freedom is given by since full pose is measured. In practice, many more measurements should be taken to offset the effect of noise in the experimental measurements. The optimisation procedure used is known as ZXSSO, and is a standard library function in the IMSL package . 2.3 Pose Measurement It is apparent from the above that a means to determine the full pose of the PUMA is required in order to perform the calibration. This method will now be described in detail. The end-effector consists of an arrangement of five precisiontooling balls as shown in Fig. 5. Consider the coordinates of the centre of each ball expressed in terms of the tool frame (Fig. 5) and the world coordinate frame, as shown in Fig. 6. The relationship between these coordinates may be written as: where Pi' is the 4 x 1 column vector of the coordinates of the ith ball expressed with respect to the world frame, P~ is the 4 x 1 column vector of the coordinates of the ith ball expressed with respect to the tool frame, and T is the 4 ? 4 homogenious transform from the world frame to the tool frame. Then may be found, and used as the measured pose in the calibration process. It is not quite that simple, however, since it is not possible to invert equation (11) to obtain T. The above process is performed for the four balls, A, B, C and D, and the positions ordered as: or in the form: Since P', T and P are all now square, the pose matrix may be obtained by inversion: In practice it may be difficult for the CMM to access four bails to determine P~ when the PUMA is placed in certain configurations. Three balls are actually measured and a fourth ball is fictitiously located according to the vector cross product: Regarding the determination of the coordinates of the centre of a ball based on measured points on its surface, no analytical procedures are available. Another numerical optimisation scheme was used for this purpose such that the penalty function: was minimised, where (u, v, w) are the coordinates of the centre of the ball to he determined, (x/, y~, z~) are the coordinates of the ith point on the surface of the ball and r is the ball diameter. In the tests performed, it was found sufficient to measure only four points (i = 4) on the surface to determine the ball centre.