0098-機(jī)械手-集裝箱波紋板焊接機(jī)器人機(jī)構(gòu)運(yùn)動(dòng)學(xué)分析及車(chē)體結(jié)構(gòu)【全套13張CAD圖+文獻(xiàn)翻譯+說(shuō)明書(shū)】

0098-機(jī)械手-集裝箱波紋板焊接機(jī)器人機(jī)構(gòu)運(yùn)動(dòng)學(xué)分析及車(chē)體結(jié)構(gòu)【全套13張CAD圖+文獻(xiàn)翻譯+說(shuō)明書(shū)】,全套13張CAD圖+文獻(xiàn)翻譯+說(shuō)明書(shū),機(jī)械手,集裝箱,波紋,焊接,機(jī)器人,機(jī)構(gòu),運(yùn)動(dòng)學(xué),分析,車(chē)體,結(jié)構(gòu),全套,13,cad,文獻(xiàn),翻譯,說(shuō)明書(shū),仿單 任務(wù)書(shū)設(shè)計(jì)(論文)題目：集裝箱波紋板焊接機(jī)器人機(jī)構(gòu)運(yùn)動(dòng)學(xué)分析及車(chē)體結(jié)構(gòu)設(shè)計(jì)畢 業(yè) 設(shè) 計(jì)（論 文）任 務(wù) 書(shū)1本畢業(yè)設(shè)計(jì)（論文）課題應(yīng)達(dá)到的目的：針對(duì)集裝箱波紋板焊接自動(dòng)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(dòng)焊專機(jī)，由于在焊接過(guò)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(jìn)而影響焊縫的質(zhì)量。該課題能有效的解決焊接過(guò)程中焊槍速度與波形夾角的問(wèn)題，使焊接速度始終與波形垂直，進(jìn)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(jì)（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(jī)器人機(jī)構(gòu)運(yùn)動(dòng)學(xué)分析及車(chē)體機(jī)構(gòu)設(shè)計(jì)，通過(guò)十字滑塊選用，進(jìn)而組成的焊接機(jī)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(wèn)題。研究?jī)?nèi)容如下：1、在廣泛調(diào)研的基礎(chǔ)上，熟悉機(jī)器人的應(yīng)用的現(xiàn)場(chǎng)環(huán)境，明確設(shè)計(jì)目標(biāo)；2、設(shè)計(jì)出該焊接機(jī)器人的機(jī)構(gòu)方案，并對(duì)其進(jìn)行運(yùn)動(dòng)學(xué)逆界，證明所選方案可行；3、設(shè)計(jì)出小車(chē)車(chē)體結(jié)構(gòu)，并在圖紙上繪制出機(jī)器人的裝配圖。應(yīng)達(dá)到的技術(shù)要求如下：1、所求焊接過(guò)渡段中的過(guò)渡運(yùn)動(dòng)能較好的銜接直線段與波內(nèi)斜邊段的運(yùn)動(dòng)；2、三自由度焊接機(jī)器人三個(gè)運(yùn)動(dòng)關(guān)節(jié)可按照一定的運(yùn)動(dòng)規(guī)律協(xié)調(diào)動(dòng)作。畢 業(yè) 設(shè) 計(jì)（論 文）任 務(wù) 書(shū)3對(duì)本畢業(yè)設(shè)計(jì)（論文）課題成果的要求包括畢業(yè)設(shè)計(jì)論文、圖表、實(shí)物樣品等： 1、相關(guān)資料的英文翻譯與文獻(xiàn)綜述； 2、焊接機(jī)器人的機(jī)構(gòu)方案；3、小車(chē)車(chē)體結(jié)構(gòu)設(shè)計(jì)及相關(guān)零件圖、裝配圖；4、畢業(yè)設(shè)計(jì)論文。畢業(yè)設(shè)計(jì)成果以設(shè)計(jì)圖樣和說(shuō)明書(shū)形式提交。要求圖樣規(guī)范，符合國(guó)家標(biāo)準(zhǔn)；說(shuō)明書(shū)層次分明、論據(jù)可靠、計(jì)算正確、圖標(biāo)規(guī)范、語(yǔ)句通順。4主要參考文獻(xiàn)：1鄭相鋒，胡小建弧焊機(jī)器人焊接區(qū)視覺(jué)信息傳感與控制技術(shù)J電焊機(jī)，20052孔宇，戴明，吳林機(jī)器人結(jié)構(gòu)光視覺(jué)三點(diǎn)焊縫定位技術(shù)J焊接學(xué)報(bào)，19973王軍波等基于CCD傳感器的球罐焊接機(jī)器人焊縫跟蹤J焊接學(xué)報(bào)，20014徐培全等基于機(jī)器人焊接的視覺(jué)傳感系統(tǒng)綜述J焊接，20055劉蘇宜，王國(guó)榮，鐘繼光視覺(jué)系統(tǒng)在機(jī)器人焊接中的應(yīng)用與展望J機(jī)械科學(xué)與技術(shù)，20056張柯等移動(dòng)焊接機(jī)器人的研究現(xiàn)狀及發(fā)展趨勢(shì)J焊接，20047王其隆弧焊過(guò)程質(zhì)量實(shí)時(shí)傳感與控制M北京：機(jī)械工業(yè)出版社，2000畢 業(yè) 設(shè) 計(jì)（論 文）任 務(wù) 書(shū)5本畢業(yè)設(shè)計(jì)（論文）課題工作進(jìn)度計(jì)劃：起 迄 日 期工 作 內(nèi) 容 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(zhǔn)備相關(guān)資料，完成資料翻譯完成文獻(xiàn)綜述，撰寫(xiě)開(kāi)題報(bào)告，熟悉AutoCAD、Pro/Engineer等繪圖軟件掌握焊接機(jī)器人的結(jié)構(gòu)設(shè)計(jì)原理，完成三自由度焊接機(jī)器人的運(yùn)動(dòng)學(xué)逆解完成車(chē)體結(jié)構(gòu)設(shè)計(jì)，畫(huà)出相關(guān)的零件圖和裝配圖撰寫(xiě)并打印設(shè)計(jì)說(shuō)明書(shū)，整理相關(guān)資料準(zhǔn)備論文答辯 外文資料翻譯(用外文寫(xiě))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。 附件1：外文資料翻譯譯文應(yīng)用坐標(biāo)測(cè)量機(jī)的機(jī)器人運(yùn)動(dòng)學(xué)姿態(tài)的標(biāo)定這篇文章報(bào)到的是用于機(jī)器人運(yùn)動(dòng)學(xué)標(biāo)定中能獲得全部姿態(tài)的操作裝置坐標(biāo)測(cè)量機(jī)（CMM）。運(yùn)動(dòng)學(xué)模型由于操作器得到發(fā)展, 它們關(guān)系到基坐標(biāo)和工件。 工件姿態(tài)是從實(shí)驗(yàn)測(cè)量中引出的討論, 同樣地是識(shí)別方法學(xué)。允許定義觀察策略的完全模擬實(shí)驗(yàn)已經(jīng)實(shí)現(xiàn)。實(shí)驗(yàn)工作的目的是描寫(xiě)參數(shù)辨認(rèn)和精確確認(rèn)。用推論原則的那方法能得到在重復(fù)時(shí)近連續(xù)地校準(zhǔn)機(jī)器人。關(guān)鍵字：機(jī)器人標(biāo)定 坐標(biāo)測(cè)量 參數(shù)辨認(rèn) 模擬學(xué)習(xí) 精確增進(jìn)1. 前言機(jī)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(shí)現(xiàn)機(jī)器手精確性，機(jī)器人可能要校準(zhǔn)也是好理解 。 在標(biāo)定過(guò)程中， 幾個(gè)連續(xù)的步驟能夠精確地識(shí)別機(jī)器人運(yùn)動(dòng)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(yùn)動(dòng)學(xué)模型和標(biāo)定過(guò)程本身是發(fā)展，和通常有標(biāo)準(zhǔn)運(yùn)動(dòng)學(xué)模型的工具實(shí)現(xiàn)的。作為結(jié)果的模型是定義基于廠商的運(yùn)動(dòng)學(xué)參數(shù)設(shè)置錯(cuò)誤量, 和識(shí)別未知的,實(shí)際的參數(shù)設(shè)置。2 機(jī)器人姿態(tài)的實(shí)驗(yàn)測(cè)量法(部分的或完成) 是拿走為了獲得從聯(lián)系到實(shí)際機(jī)器人的參數(shù)設(shè)置數(shù)據(jù)。3 實(shí)際的運(yùn)動(dòng)學(xué)參數(shù)識(shí)別是系統(tǒng)地改變參數(shù)設(shè)置和減少在模型階段錯(cuò)誤量的定義。一個(gè)接近完成辨認(rèn)由分析不同中間姿態(tài)變量P和運(yùn)動(dòng)學(xué)參數(shù)K的微分關(guān)系決定：于是等價(jià)轉(zhuǎn)化得：兩者擇一, 問(wèn)題可以看成為多維的優(yōu)化問(wèn)題，這是為了減少一些定義的錯(cuò)誤功能到零點(diǎn)，運(yùn)動(dòng)學(xué)參數(shù)設(shè)置被改變。這是標(biāo)準(zhǔn)優(yōu)化問(wèn)題和可能解決用的眾所周知的 方法。4 最后一步是機(jī)械手控制中的機(jī)器人運(yùn)動(dòng)學(xué)識(shí)別和在學(xué)習(xí)之下的硬件系統(tǒng)的詳細(xì)資料。包含實(shí)驗(yàn)數(shù)據(jù)的這張紙用于標(biāo)度過(guò)程。 可獲得的幾個(gè)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(shí)間。這樣的技術(shù)包括使用可視化的和自動(dòng)化機(jī)械 ，伺服控制激光干涉計(jì)，有關(guān)聲音的傳感器和視覺(jué)傳感器 。理想測(cè)量系統(tǒng)將獲得操作器的全部姿態(tài)(位置和方向)，因?yàn)檫@將合并機(jī)械臂各個(gè)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(biāo)度過(guò)程到進(jìn)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(cè)量的，雖然進(jìn)行幾個(gè)中間測(cè)量，是為了獲得姿態(tài)。測(cè)量姿態(tài)使用裝置是坐標(biāo)測(cè)量機(jī)(CMM),它是三軸的，棱鏡測(cè)量系統(tǒng)達(dá)到0.01毫米的精確。機(jī)器人操作器是能校準(zhǔn)的，PUMA 560，放置接近于CMM，特殊的操作裝置能到達(dá)邊緣。圖1顯示了系統(tǒng)不同部分安排。在這部分運(yùn)動(dòng)學(xué)模型將是發(fā)展, 解釋姿態(tài)估算法，和參數(shù)辨認(rèn)方法。2.1 運(yùn)動(dòng)學(xué)的參數(shù)在這部分，操作器的基本運(yùn)動(dòng)學(xué)結(jié)構(gòu)將被規(guī)定，它關(guān)系到完全坐標(biāo)系統(tǒng)的討論, 和終點(diǎn)模型。從這些模型，用于可能的技術(shù)的運(yùn)動(dòng)學(xué)參數(shù)的識(shí)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(xiě)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(jì)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(shuō)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(yùn)動(dòng)學(xué)中需要從一個(gè)坐標(biāo)到另一個(gè)坐標(biāo)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(guò)四個(gè)基本變化操作實(shí)現(xiàn)坐標(biāo)系n-1到結(jié)構(gòu)坐標(biāo)系n的變化。只有需要找到與前一個(gè)的關(guān)系的四個(gè)變化是必需的，在那個(gè)時(shí)候連續(xù)的軸是不平行的，定義為零點(diǎn)。當(dāng)應(yīng)用于一個(gè)結(jié)構(gòu)到下一個(gè)結(jié)構(gòu)的等價(jià)變化坐標(biāo)系與更改Denavit-Hartenberg系相一致時(shí)，它們將被書(shū)寫(xiě)成矩陣元素實(shí)現(xiàn)運(yùn)動(dòng)學(xué)參數(shù)功能的矩陣形狀。這些參數(shù)是變化的簡(jiǎn)單變量：關(guān)節(jié)角，連桿偏置, 連桿長(zhǎng)度，扭角，矩陣通常表示如下：對(duì)于多連接的, 例如機(jī)械操作臂,各自連續(xù)的鏈環(huán)和兩者瞬間的位置描寫(xiě)在前一個(gè)矩陣變化中。這種變化從底部鏈環(huán)開(kāi)始到第n鏈環(huán)因此關(guān)系如下：圖3表示出PUMA機(jī)器人在Denavit-Hartenberg系中每一連桿，完全坐標(biāo)系和工具結(jié)構(gòu)。變化從世界坐標(biāo)系到機(jī)器人底部結(jié)構(gòu)需要仔細(xì)考慮過(guò)，因?yàn)闈撛诘膮?shù)取決于被選擇的改變類型?？紤]到圖4,世界坐標(biāo)，在D-H系中定義的從世界坐標(biāo)到機(jī)器人基坐標(biāo)，坐標(biāo)是PUMA機(jī)器人定義的基坐標(biāo)和機(jī)器人第二個(gè)D-H結(jié)構(gòu)中坐標(biāo)。我們感興趣的是從世界坐標(biāo)到必需的最小的參數(shù)數(shù)量。實(shí)現(xiàn)這種變化有兩種路徑：路徑1，從到D-H變化包括四個(gè)參數(shù)，接著從到的變化將牽連二個(gè)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(gè)參數(shù)其中和兩個(gè)參數(shù)是關(guān)于軸Z0因此不能獨(dú)立地識(shí)別， 和是沿著軸Z0因此也不能是獨(dú)立地識(shí)別。因此，用這路徑它需要從世界坐標(biāo)到PUMA機(jī)器人的第一個(gè)坐標(biāo)有八個(gè)獨(dú)立的運(yùn)動(dòng)學(xué)參數(shù)。路徑2，同樣地二中擇一，從世界坐標(biāo)到底部結(jié)構(gòu)坐標(biāo)的變化可以是直接定義。因此坐標(biāo)變換需要六個(gè)參數(shù)，如Euler形式：下面是從到DH變化中的四個(gè)參數(shù)，但與相關(guān)聯(lián)，與相關(guān)聯(lián)，減少成兩個(gè)參數(shù)。很顯然這種路徑和路徑1一樣需要八個(gè)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(biāo)系到PUMA機(jī)器人的第二結(jié)構(gòu)的移動(dòng)中。在這工作中，選擇路徑2。工具改變引起需要六個(gè)特殊參數(shù)的改變的Euler形式：用于運(yùn)動(dòng)學(xué)模型的參數(shù)總數(shù)變成30，他們定義于表12.2 辨認(rèn)方法學(xué)運(yùn)動(dòng)學(xué)的參數(shù)辨認(rèn)將是進(jìn)行多維的消去過(guò)程, 因此避免了雅可比系統(tǒng)的標(biāo)定，過(guò)程如下：1. 首先假設(shè)運(yùn)動(dòng)學(xué)的參數(shù), 例如標(biāo)準(zhǔn)設(shè)置。2. 為選擇任意關(guān)節(jié)角的設(shè)置。3. 計(jì)算PUMA機(jī)器人末端操作器。4. 測(cè)量PUMA機(jī)器人末端操作器的位姿如關(guān)節(jié)角，通常標(biāo)準(zhǔn)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(dá)到標(biāo)準(zhǔn)的位姿，在整齊的方式更改運(yùn)動(dòng)學(xué)的參數(shù)。這個(gè)過(guò)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(cè)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(shí)別的運(yùn)動(dòng)學(xué)參數(shù)的數(shù)量N是測(cè)量位姿的數(shù)Dr是測(cè)量過(guò)程中自由度的數(shù)量文章中，給定了自由度的數(shù)量，贈(zèng)值為因此全部位姿是測(cè)量的。在實(shí)踐中，更多的測(cè)量應(yīng)該是在實(shí)驗(yàn)測(cè)量法去掉補(bǔ)償結(jié)果。優(yōu)化程序使用命名為ZXSSO，和標(biāo)準(zhǔn)庫(kù)功能的IMSL。2.3 位姿測(cè)量法顯然它是從上面的方法確定PUMA機(jī)器人全部位姿是必需的為了實(shí)現(xiàn)標(biāo)定。這種方法現(xiàn)在將詳細(xì)地描寫(xiě)。如圖5所示，末端操作器由五個(gè)確定的工具組成。 考慮到借助于工具坐標(biāo)和世界坐標(biāo)中間各個(gè)坐標(biāo)的形式，如圖6這些坐標(biāo)的關(guān)系如下：是關(guān)于世界坐標(biāo)結(jié)構(gòu)的第i個(gè)球的4x1列向量坐標(biāo), Pi是關(guān)于工具坐標(biāo)結(jié)構(gòu)第i個(gè)球的4x1坐標(biāo)的列向量, T是從世界坐標(biāo)結(jié)構(gòu)到工具坐標(biāo)結(jié)構(gòu)變化的4x4矩陣。設(shè)定Pi，測(cè)量出，然后算出T，使用于在標(biāo)定過(guò)程的位姿的測(cè)量。它是不會(huì)很簡(jiǎn)單，但是不可能由等式(11)反求出T。上面的過(guò)程由四個(gè)球A, B, C和D來(lái)實(shí)現(xiàn)，如下：或?yàn)橛捎赑, T和P全部相符合，反解求的位姿矩陣在實(shí)踐中當(dāng)PUMA機(jī)器人放置在確定的位置上，對(duì)于CMM由四個(gè)球決定Pi是困難的。準(zhǔn)確的測(cè)量三個(gè)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(biāo)的是基于球表面點(diǎn)的測(cè)量,沒(méi)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(gè)球表面點(diǎn)的坐標(biāo)且是球的半徑。在測(cè)試過(guò)程中，發(fā)現(xiàn)只測(cè)量四個(gè)表面上的點(diǎn)來(lái)確定中心點(diǎn)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 kinematiccalibration of a robot manipulator using a coordinate measuringmachine (CMM) which is able to obtain the full pose ofthe end-effector. A kinematic model is developed for themanipulator, its relationship to the world coordinate frame andthe tool. The derivation of the tool pose from experimentalmeasurements is discussed, as is the identification methodology.A complete simulation of the experiment is performed, allowingthe observation strategy to be defined. The experimental workis described together with the parameter identification andaccuracy verification. The principal conclusion is that themethod is able to calibrate the robot successfully, with aresulting accuracy approaching that of its repeatability.Keywords: Robot calibration; Coordinate measurement; Parameter identification; Simulation study; Accuracy enhancement1. Introduction It is well known that robot manipulators typically havereasonable repeatability (0.3 ram), yet exhibit poor accuracy(10.0 mm). The process by which robots may be calibratedin order to achieve accuracies approaching that of themanipulator is also well understood . In the calibrationprocess, several sequential steps enable the precise kinematicparameters of the manipulator to be identified, leading toimproved accuracy. These steps may be described as follows:1. A kinematic model of the manipulator and the calibrationprocess itself is developed and is usually accomplished withstandard kinematic modelling tools. The resulting modelis used to define an error quantity based on a nominal(manufacturers) kinematic parameter set, and an unknown,actual parameter set which is to be identified.2. Experimental measurements of the robot pose (partial orcomplete) are taken in order to obtain data relating to theactual parameter set for the robot.3.The actual kinematic parameters are identified by systematicallychanging the nominal parameter set so as to reducethe error quantity defined in the modelling phase. Oneapproach to achieving this identification is determiningthe analytical differential relationship between the posevariables P and the kinematic parameters K in the formof a Jacobian, and then inverting the equation to calculate the deviation ofthe kinematic parameters from their nominal valuesAlternatively, the problem can be viewed as a multidimensionaloptimisation task, in which the kinematic parameterset is changed in order to reduce some defined error functionto zero. This is a standard optimisation problem and maybe solved using well-known methods.4. The final step involves the incorporation of the identifiedkinematic parameters in the controller of the robot arm,the details of which are rather specific to the hardware ofthe system under study. This paper addresses the issue of gathering the experimentaldata used in the calibration process. Several methods areavailable to perform this task, although they vary in complexity,cost and the time taken to acquire the data. Examples ofsuch techniques include the use of visual and automatictheodolites, servocontrolled laser interferometers ,acoustic sensors and vidual sensors . An ideal measuringsystem would acquire the full pose of the manipulator (positionand orientation), because this would incorporate the maximuminformation for each position of the arm. All of the methodsmentioned above use only the partial pose, requiring moredata to be taken for the calibration process to proceed.2. Theory In the method described in this paper, for each position inwhich the manipulator is placed, the full pose is measured,although several intermediate measurements have to be takenin order to arrive at the pose. The device used for the posemeasurement is a coordinate-measuring machine (CMM),which is a three-axis, prismatic measuring system with aquoted accuracy of 0.01 ram. The robot manipulator to becalibrated, a PUMA 560, is placed close to the CMM, and aspecial end-effector is attached to the flange. Fig. 1 showsthe arrangement of the various parts of the system. In thissection the kinematic model will be developed, the poseestimation algorithms explained, and the parameter identificationmethodology outlined.2.1 Kinematic Parameters In this section, the basic kinematic structure of the manipulatorwill be specified, its relation to a user-defined world coordinatesystem discussed, and the end-point toil modelled. From thesemodels, the kinematic parameters which may be identifiedusing the proposed technique will be specified, and a methodfor determining those parameters described.The fundamental modelling tool used to describe the spatialrelationship between the various objects and locations in themanipulator 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. Asshown in Fig. 2, this method places a coordinate frame oneach object or manipulator link of interest, and the kinematicsare defined by the homogeneous transformation required tochange one coordinate frame into the next. This transformationtakes the familiar form The above equation may be interpreted as a means totransform frame n-1 into frame n by means of four out ofthe five operations indicated. It is known that only fourtransformations are needed to locate a coordinate frame withrespect to the previous one. When consecutive axes are notparallel, the value of/3. is defined to be zero, while for thecase when consecutive axes are parallel, d. is the variablechosen to be zero. When coordinate frames are placed in conformance withthe modified Denavit-Hartenberg method, the transformationsgiven in the above equation will apply to all transforms ofone frame into the next, and these may be written in ageneric matrix form, where the elements of the matrix arefunctions of the kinematic parameters. These parameters aresimply the variables of the transformations: the joint angle0., the common normal offset d., the link length a., the angleof twist a., and the angle /3. The matrix form is usuallyexpressed as follows: For a serial linkage, such as a robot manipulator, a coordinateframe is attached to each consecutive link so that both theinstantaneous position together with the invariant geometryare described by the previous matrix transformation. Thetransformation from the base link to the nth link will thereforebe given byFig. 3 shows the PUMA manipulator with theDenavit-Hartenberg frames attached to each link, togetherwith world coordinate frame and a tool frame. The transformationfrom the world frame to the base frame of themanipulator needs to be considered carefully, since there arepotential parameter dependencies if certain types of transformsare chosen. Consider Fig. 4, which shows the world framexw, y, z, the frame Xo, Yo, z0 which is defined by a DHtransform from the world frame to the first joint axis ofthe manipulator, frame Xb, Yb, Zb, which is the PUMAmanufacturers defined base frame, and frame xl, Yl, zl whichis the second DH frame of the manipulator. We are interestedin determining the minimum number of parameters requiredto move from the world frame to the frame x, Yl, z. Thereare two transformation paths that will accomplish this goal:Path 1: A DH transform from x, y, z, to x0, Yo, zoinvolving four parameters, followed by another transformfrom xo, Yo, z0 to Xb, Yb, Zb which will involve only twoparameters b and d in the transformFinally, another DH transform from xb, Yb, Zb to Xt, y, Zwhich involves four parameters except that A01 and 4 areboth about the axis zo and cannot therefore be identifiedindependently, and Adl and d are both along the axis zo andalso cannot be identified independently. It requires, therefore,only eight independent kinematic parameters to go from theworld frame to the first frame of the PUMA using this path.Path 2: As an alternative, a transform may be defined directlyfrom the world frame to the base frame Xb, Yb, Zb. Since thisis a frame-to-frame transform it requires six parameters, suchas the Euler form:The following DH transform from xb, Yb, zb tO Xl, Yl, zlwould involve four parameters, but A0 may be resolved into4, 0b, , and Ad resolved into Pxb, Pyb, Pzb, reducing theparameter count to two. It is seen that this path also requireseight parameters as in path i, but a different set.Either of the above methods may be used to move fromthe world frame to the second frame of the PUMA. In thiswork, the second path is chosen. The tool transform is anEuler transform which requires the specification of sixparameters:The total number of parameters used in the kinematic modelbecomes 30, and their nominal values are defined in Table 1.2.2 Identification Methodology The kinematic parameter identification will be performed asa multidimensional minimisation process, since this avoids thecalculation of the system Jacobian. The process is as follows:1. Begin with a guess set of kinematic parameters, such asthe 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 forthe same set of joint angles. In general, the measured andpredicted pose will be different.5. Modify the kinematic parameters in an orderly manner inorder to best fit (in a least-squares sense) the measuredpose to the predicted pose. The process is applied not to a single set of joint angles butto a number of joint angles. The total number of joint anglesets required, which also equals the number of physicalmeasurement made, must satisfyKp is the number of kinematic parameters to be identifiedN is the number of measurements (poses) takenDr represents the number of degrees of freedom present ineach measurement. In the system described in this paper, the number of degreesof freedom is given bysince full pose is measured. In practice, many more measurementsshould be taken to offset the effect of noise in theexperimental measurements. The optimisation procedure usedis known as ZXSSO, and is a standard library function in theIMSL package .2.3 Pose Measurement It is apparent from the above that a means to determine thefull pose of the PUMA is required in order to perform thecalibration. This method will now be described in detail. Theend-effector consists of an arrangement of five precisiontoolingballs as shown in Fig. 5. Consider the coordinates ofthe 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 writtenas:where Pi is the 4 x 1 column vector of the coordinates ofthe ith ball expressed with respect to the world frame, P isthe 4 x 1 column vector of the coordinates of the ith ballexpressed with respect to the tool frame, and T is the 4 4homogenious transform from the world frame to the toolframe. Then may be found, and used as the measured pose in thecalibration process. It is not quite that simple, however, sinceit is not possible to invert equation (11) to obtain T. Theabove process is performed for the four balls, A, B, C andD, and the positions ordered as:or in the form:Since P, T and P are all now square, the pose matrix maybe obtained by inversion:In practice it may be difficult for the CMM to access fourbails to determine P when the PUMA is placed in certainconfigurations. Three balls are actually measured and a fourthball is fictitiously located according to the vector cross product:Regarding the determination of the coordinates of thecentre of a ball based on measured points on its surface,no analytical procedures are available. Another numericaloptimisation scheme was used for this purpose such that thepenalty function:was minimised, where (u, v, w) are the coordinates of thecentre of the ball to he determined, (x/, y, z) are thecoordinates of the ith point on the surface of the ball and ris the ball diameter. In the tests performed, it was foundsufficient to measure only four points (i = 4) on the surfaceto determine the ball centre.