立式高速銑削加工中心縱向進(jìn)給機(jī)構(gòu)設(shè)計(jì)5張CAD圖

立式高速銑削加工中心縱向進(jìn)給機(jī)構(gòu)設(shè)計(jì)5張CAD圖,立式,高速,銑削,加工,中心,縱向,進(jìn)給,機(jī)構(gòu),設(shè)計(jì),CAD 附錄一： 減少加工誤差的三軸機(jī)床機(jī)械測(cè)量和誤差補(bǔ)償系統(tǒng) 機(jī)械工程研究生院,延世大學(xué),首爾,韓國(guó) 機(jī)械工程系,延世大學(xué),首爾,韓國(guó) 文摘： 提出一個(gè)方法來減少加工誤差的三軸機(jī)床通過實(shí)現(xiàn)一個(gè)機(jī)器測(cè)量觸摸探針。觸摸探針探測(cè)錯(cuò)誤和機(jī)床的定位錯(cuò)誤,不可避免地包含在測(cè)量數(shù)據(jù),彌補(bǔ)獲得真實(shí)的加工錯(cuò)誤的重復(fù)加工過程。 定位錯(cuò)誤的工具/探針尖被逼近誤差建模組件作為多項(xiàng)式函數(shù)和考慮抵制錯(cuò)誤的影響。 來估計(jì)未知模型參數(shù)、多維數(shù)組工件組成的八個(gè)數(shù)據(jù)集提出了 CMM 和校準(zhǔn)。仿真結(jié)果和驗(yàn)證實(shí)驗(yàn)表明,測(cè)量和預(yù)測(cè)定位錯(cuò)誤同意不到 10 米之內(nèi)所有軸。 一個(gè)簡(jiǎn)單的塊的實(shí)際切削試驗(yàn)和二維曲線表明,加工錯(cuò)誤減少到后 10 米 內(nèi) 的 錯(cuò) 誤 補(bǔ) 償 。 2004 愛 思 唯 爾 出 版 的 帳 面 價(jià) 值 。 關(guān)鍵詞:機(jī)械測(cè)量(介質(zhì));碰探頭,立方體數(shù)組工件;誤差補(bǔ)償 1、介紹 在傳統(tǒng)制造工藝,檢驗(yàn)部分完成了獨(dú)立的測(cè)量?jī)x器,如坐標(biāo)測(cè)量機(jī)(CMM), 通常位于一個(gè)單獨(dú)的房間除了機(jī)床。這就增加了整體生產(chǎn)成本和時(shí)間獲得最終產(chǎn)品, 和瓶頸現(xiàn)象可能是由于產(chǎn)品停滯由于加工之間的時(shí)間差和審查過程的柔性制造系統(tǒng)。此外,很難轉(zhuǎn)移、夾具和測(cè)量復(fù)雜的大型零件[1]。 為了克服這些問題,一個(gè)在機(jī)器測(cè)量(石)系統(tǒng)見圖 1，是使用一個(gè)商業(yè)實(shí)現(xiàn)觸摸探針(從英國(guó) MP10 Inc .)。 觸摸探針是相對(duì)便宜和易于使用的配件,可以實(shí)現(xiàn)顯著減少生產(chǎn)時(shí)間和成本,廣泛用于過程改進(jìn)自動(dòng)化和加速處理一部分,甚至消除一部分錯(cuò)誤的過程。光模塊的系統(tǒng)由探針(OMP)和光學(xué)機(jī)接口(OMI)。OMP,位于探測(cè)頭和柄之間,收到機(jī)器控制信號(hào)和傳輸探頭信號(hào)。探測(cè)器和 OMI 之間的通信是通過光傳輸系統(tǒng),而使用 rs - 232 串行通信傳輸測(cè)量程序(宏程序)CNC 控制器和接收的測(cè)量數(shù)據(jù)進(jìn)一步分析使用個(gè)人電腦。 圖 2 顯示了本研究的整體工作流程提高機(jī)械加工精度的測(cè)量和誤差補(bǔ)償系統(tǒng)。數(shù)控使用部分模型生成的數(shù)據(jù)被用來喂養(yǎng) CNC 控制器用于加工第一步。加工過程結(jié)束后,觸摸探針換成了刀具開始測(cè)量加工表面的法線方向。自從接觸探頭沿著錯(cuò)誤的措施部分機(jī)床軸,測(cè)量的數(shù)據(jù)不可避免地包括探測(cè)錯(cuò)誤源于觸摸探針的結(jié)構(gòu)特點(diǎn),和定位錯(cuò)誤源于不準(zhǔn)確的軸運(yùn)動(dòng)的機(jī)床。這些錯(cuò)誤應(yīng)該取消從測(cè)量數(shù)據(jù)來獲得真實(shí)的加工誤差。如果真正的加工誤差大于給定的公差,新刀具軌跡生成使用下一步加工的誤差補(bǔ)償算法。加工和機(jī)械測(cè)量過程不斷重復(fù),直到所需的部分公差,導(dǎo)致閉環(huán)加工系統(tǒng)[2]。 提出一個(gè)方法來快速評(píng)估機(jī)床的定位錯(cuò)誤使用一個(gè)新的數(shù)組工件誤差模型和多維數(shù)據(jù)集。誤差模型是由近似誤差組件出現(xiàn)在體積誤差模型和多項(xiàng)式函數(shù)。前后誤差模型分解為模型根據(jù)機(jī)床的軸的運(yùn)動(dòng)方向,因?yàn)榉磳?duì)錯(cuò)誤影響機(jī)器測(cè)量數(shù)據(jù)。系數(shù)來確定未知的模型,一個(gè)多維數(shù)組工件組成的八個(gè)數(shù)據(jù)集提出了 CMM 和校準(zhǔn)。在立方體頂點(diǎn)定位錯(cuò)誤的仿真結(jié)果顯示,估計(jì)錯(cuò)誤也同意所有軸的測(cè)量誤差在向前和向后的方向。計(jì)是用于驗(yàn)證一步建議誤差模型。最后,一個(gè)簡(jiǎn)單的塊和二維曲線的加 工測(cè)試執(zhí)行,在一個(gè)基于線分割算法的誤差補(bǔ)償方法應(yīng)用于減少加工錯(cuò)誤。它可以得出的結(jié)論是,加工錯(cuò)誤減少到誤差補(bǔ)償后 10 米內(nèi)。 通 訊 作 者 電 子 郵 件 地 址 :feel2@korea.com( 大 通 ), bkmin@yonsei.ac?；?雷克南(Min)的手段,sjlee@yonsei.ac?；?雷克南(S.J. Lee) 。 0924 - 0136 / $ - 見 前 頁 ?2004 愛 思 唯 爾 出 版 有 限 責(zé) 任 公 司doi:10.1016 / j.jmatprotec.2004.04.402 2、表征探測(cè)錯(cuò)誤和定位錯(cuò)誤 2.1 探測(cè)錯(cuò)誤 在觸摸探針,機(jī)械結(jié)構(gòu)支持手寫筆作為電觸發(fā)開關(guān),當(dāng)筆流離失所。這個(gè)結(jié)果與分裂的探針天線波束的控制結(jié)構(gòu)反映出三角形接觸探頭內(nèi)的機(jī)械結(jié)構(gòu) [3]。因?yàn)檫@些探測(cè)錯(cuò)誤影響到測(cè)量數(shù)據(jù)根據(jù)不同的調(diào)查方法方向,他們必須得到補(bǔ)償,然后再執(zhí)行實(shí)際的測(cè)量。圖3 顯示了通過測(cè)量獲得的探測(cè)錯(cuò)誤一個(gè)精確的環(huán)規(guī)直徑為29.998 毫米。球針長(zhǎng)度 50 mm 和探針半徑為 1 毫米。探測(cè)誤差的大小取決于針長(zhǎng)度和方向的調(diào)查。誤差補(bǔ)償后,探測(cè)錯(cuò)誤減少到 5 米之內(nèi),同一訂單的機(jī)床的可重復(fù)性。 2.2 機(jī)床誤差的數(shù)學(xué)公式 機(jī)床誤差傳播到機(jī)器測(cè)量數(shù)據(jù),自從接觸探頭沿著錯(cuò)誤的措施部分機(jī)床軸。所以, 這些錯(cuò)誤應(yīng)該在機(jī)器的識(shí)別和消除測(cè)量數(shù)據(jù)獲取項(xiàng)目下一步加工過程的真實(shí)加工錯(cuò)誤。確定工作空間內(nèi)的任何位置的定位錯(cuò)誤,一般齊次變換矩陣(HTM),這代表剛體坐標(biāo)系統(tǒng)的坐標(biāo)變換幀的參考坐標(biāo)系統(tǒng)[4]。增加移動(dòng)元素及其誤差矩陣的 htm 先后從參考坐標(biāo)系到實(shí)際工具坐標(biāo)系位置得到的理想位置和機(jī)床的所有錯(cuò)誤組件。圖 4 顯示了坐標(biāo)系的三軸機(jī)床用于 thisresearch,和由此產(chǎn)生的定位誤差來源于以下方程: 這里,δii(i = x,y,z)表示線性錯(cuò)誤面前是沿著軸,δij(i,j = x,y,z 和我= j)第 i 個(gè)軸方向的直線度誤差沿著 jth 軸時(shí),εij 角錯(cuò)誤在第 i 個(gè)軸滑動(dòng)沿著 jth 軸移 動(dòng),Sij 之間的垂直度誤差對(duì)應(yīng)的軸。和人工智能,bi,ci 原點(diǎn)偏移量從(我?1)屆第 i 個(gè)坐標(biāo)系坐標(biāo)系統(tǒng),和 L 的理想工具沿著 z 軸長(zhǎng)度(表 1)。 表 1 預(yù)測(cè)工具的定位/探針針尖在工作區(qū)使用 Eq。(1),21 個(gè)錯(cuò)誤組件的測(cè)量數(shù)據(jù)應(yīng)該是必需的。激光干涉儀系統(tǒng)被廣泛用于測(cè)量這些錯(cuò)誤與精度高,但它需要長(zhǎng)時(shí)間校準(zhǔn)時(shí)間和成本[5]。評(píng)估定位錯(cuò)誤更加快速和簡(jiǎn)單的方式,體積誤差模型參數(shù)使用圖 4。列遍歷立式加工中心的坐標(biāo)系統(tǒng)。確定模型參數(shù),可以簡(jiǎn)單地使用一個(gè)觸摸探針和工件。獲得參數(shù)誤差模型、線性和角度錯(cuò)誤假設(shè)作為第一和二階多項(xiàng)式函數(shù)[6]。直線度誤差推導(dǎo)通過集成角錯(cuò)誤和方形錯(cuò)誤被視為常數(shù)無論軸位置。替換組件到體積誤差模型,近似參數(shù)誤差模型得到矩陣方程的形式: EFWD 3×1 誤差向量,3×15 標(biāo)量矩陣 B、p 15×1 系數(shù)向量的未知參數(shù)。誤差向量的下標(biāo)表示誤差模型是適用于軸方向的移動(dòng),所有錯(cuò)誤組件被設(shè)置為 0 的相應(yīng)軸的原點(diǎn)。模型參數(shù)向量 p 可以很容易地使用最小二乘估計(jì)量決定的。 自從接觸探頭沿機(jī)床軸的措施部分,反對(duì)錯(cuò)誤的錯(cuò)誤的組件移動(dòng)軸影響測(cè)量數(shù)據(jù)除了在給定位置定位錯(cuò)誤,因此他們必須被包括在誤差模型。從使用激光干涉儀系統(tǒng)的初步實(shí)驗(yàn)結(jié)果,反對(duì)錯(cuò)誤假定常數(shù)無論軸位置[7]。代替近似錯(cuò)誤組件包括反彈到之前的體積誤差模型和矩陣形式改寫,向后方向的誤差模型推導(dǎo)出如下： E FWD 3×1 誤差向量,3×15 標(biāo)量矩陣 B、p 15×1 系數(shù)向量的未知參數(shù)。誤差向量的下標(biāo)表示誤差模型是適用于軸方向的移動(dòng),所有錯(cuò)誤組件被設(shè)置為 0 的相應(yīng)軸的原點(diǎn)。模型參數(shù)向量 p 可以很容易地使用最小二乘估計(jì)量決定的。 p = (BTB)?1BTEFWD (3) 2060 自從接觸探頭沿機(jī)床軸的措施部分,反對(duì)錯(cuò)誤的錯(cuò)誤的組件移動(dòng)軸影響測(cè)量數(shù)據(jù)除了在給定位置定位錯(cuò)誤,因此他們必須被包括在誤差模型。從使用激光干涉儀系統(tǒng)的初步實(shí)驗(yàn)結(jié)果,反對(duì)錯(cuò)誤假定常數(shù)無論軸位置[7]。代替近似錯(cuò)誤組件包括反彈到之前的體積誤差模型和重寫. 大通崔 et al。/材料處理技術(shù)雜志》155 - 156(2004)2056 - 2004 EBWD 在哪 3×1 誤差向量方向向后,EFWD 3×1 的錯(cuò)誤方向的向量。大通崔 et al。/材料處理技術(shù)雜志》155 - 156(2004)2056 - 2004 EBWD 在哪 3×1 誤差向量方向向后,EFWD 3×1 的錯(cuò)誤方向的向量(2)),F×18 標(biāo)量矩陣 h 18×1 的系數(shù)向量確定的組件是錯(cuò)誤的反應(yīng)錯(cuò)誤組件。注意,通過添加錯(cuò)誤源于落后的錯(cuò)誤得到反彈錯(cuò)誤方向的錯(cuò)誤。模型參數(shù)向量 h 可以估計(jì)同樣像以前一樣: h = (FTF)?1FT{EBWD ? EFWD} (5) 3 模型參數(shù)估計(jì)和仿真結(jié)果 3.1 多維數(shù)組的工件 確定模型參數(shù)向量方程式的 p,h。(5),八個(gè)立方體組成的立方體數(shù)組工件如圖 5 所示(一個(gè))提出,使測(cè)量定位錯(cuò)誤都向前和向后的方向[7]。工件與坐標(biāo)校準(zhǔn)。 圖 5。多維數(shù)組工件和機(jī)器測(cè)量。 《材料處理技術(shù)雜志》155 - 156(2004)2056 - 2004 2061 2061 測(cè)量機(jī),然后安裝在機(jī)床上的桌子與觸摸探針測(cè)量右側(cè)的圖 5 所示(b)。CMM 的區(qū)別在立方體頂點(diǎn)數(shù)據(jù)和介質(zhì)數(shù)據(jù)用于生成的錯(cuò)誤矢量 EFWD 和 EBWD 向前和向后誤差模型,分別。誤差向量和名義立方體頂點(diǎn)的位置在機(jī)器坐標(biāo)系是用來確定模型參數(shù)向量。 3.2 模擬 估計(jì)模型參數(shù),定位錯(cuò)誤在立方體角落預(yù)測(cè),并與實(shí)測(cè)的錯(cuò)誤。無花果。6 和 7 比較了模擬定位誤差與測(cè)量誤差都向前和向后 x 坐標(biāo)軸和 y 坐標(biāo)軸的方向,分別。數(shù)據(jù)的二次和三次錯(cuò)誤模型意味著錯(cuò)誤組件與 fistand 近似二階多項(xiàng)式函數(shù),分別。 圖 6。模擬和測(cè)量軸的定位錯(cuò)誤。 圖 7。模擬和測(cè)量定位錯(cuò)誤的 y 軸。 3.3 使用步驟計(jì)模型驗(yàn)證 一步計(jì)10 毫米的名義塊大小和間距20 毫米的左邊圖8 所示(一個(gè))是用于驗(yàn)證誤差模型。它是安裝在機(jī)床表和測(cè)量都向前和向后的方向。 2062 《材料處理技術(shù)雜志》155 - 156(2004)2056 - 2064 圖 8 使用步驟計(jì)模型驗(yàn)證 圖 9 幾何部分用于加工實(shí)驗(yàn) 4 加工實(shí)驗(yàn) 4.1 部分幾何和誤差補(bǔ)償方案 機(jī)器測(cè)量系統(tǒng)應(yīng)用于加工測(cè)試一個(gè)簡(jiǎn)單的塊組成的廣場(chǎng)和鉆石的特性和二維曲線如圖 10 所示。 圖 10 一個(gè)簡(jiǎn)單的塊比較加工錯(cuò)誤 大通崔 et al。/材料處理技術(shù)雜志》155 - 156(2004)2056 - 2004 2063 圖 9 加工完成第一步后,刀具被替換為一個(gè)觸摸探針,用于衡量的機(jī)械加工面等距的計(jì)量點(diǎn)。計(jì)量點(diǎn)的探測(cè)誤差和定位誤差從測(cè)量數(shù)據(jù)得到真正消除加工誤差考慮 探測(cè)器接近角。注意,調(diào)查的方法是不斷沿著角兩邊的廣場(chǎng)和鉆石的特性,而測(cè)量方向不斷變化以及二維曲線。如果真正的加工誤差大于指定的公差,新刀具軌跡生成的第二次加工插值補(bǔ)償點(diǎn)測(cè)量的點(diǎn)。補(bǔ)償點(diǎn)是由添加真實(shí)加工反對(duì)派方向[8]中的錯(cuò)誤。第二次加工后使用 圖 11 比較二維曲線的加工錯(cuò)誤 真正的加工錯(cuò)誤。如果真正的加工誤差小于公差,這個(gè)過程完成后,得到最終的產(chǎn)品。否則,部分再次測(cè)量和加工誤差補(bǔ)償精度得到迭代,直到所需的部分。 4.2 加工結(jié)果 圖 10 顯示了加工誤差測(cè)量與加工第一步后觸摸探針第二次加工和測(cè)量完成后, 一部分是在 CMM 測(cè)量在同一測(cè)量與機(jī)器測(cè)量數(shù)據(jù)點(diǎn)比較。 5 結(jié)論 提出一個(gè)在機(jī)器測(cè)量和誤差補(bǔ)償系統(tǒng),減少使用觸摸探針加工錯(cuò)誤。真實(shí)加工錯(cuò)誤決心通過消除接觸探頭的探測(cè)錯(cuò)誤和機(jī)床的定位錯(cuò)誤。定位錯(cuò)誤考慮抵制錯(cuò)誤的影響嗎錯(cuò)誤的組件。提出了多維數(shù)組工件前后確定的模型參數(shù)誤差模型。仿真結(jié)果表明,該預(yù)測(cè)定位錯(cuò)誤同意所有軸的測(cè)量錯(cuò)誤該機(jī)器測(cè)量系統(tǒng)可以擴(kuò)展傳統(tǒng)機(jī)床的測(cè)量機(jī)精度檢驗(yàn)和改進(jìn)部分。 參考文獻(xiàn)： [1] 薩達(dá)姆政權(quán)金、D.H. Kim j .侯爾的機(jī)器上測(cè)量系統(tǒng)。Soc。摘要。Eng。18(6)(2001) 這 9 到 18。 [2] 薩達(dá)姆政權(quán)書釘、孫 J.W.林亭汝榮格,閉環(huán)方法減少總加工誤差:實(shí)驗(yàn)和分析,反式。NAMRI /中小企業(yè) 15(1997)311 - 316。 [3]J.A.博世,坐標(biāo)測(cè)量機(jī)和系統(tǒng),馬塞爾·德克爾公司,紐約,1995 年。 [4]交流可以用 Y.M. Ertekin,機(jī)床誤差模型的推導(dǎo)和誤差補(bǔ)償過程三軸垂直的加工2064 《料處理技術(shù)雜志》155 - 156(2004)2056 - 2064 中心使用剛體運(yùn)動(dòng)學(xué),Int,j·馬赫。工具 Manuf。40(2000)1199 - 1213。[5]交流可以用 Y.M. Ertekin,立式加工中心精度特性使用激光干涉儀,Proc。ASPE 18(1998)506 - 511。 [6]j .備忘錄 C.R. Liu 方法提高數(shù)控加工工具的準(zhǔn)確性在機(jī)器檢查,j . Manuf 系統(tǒng)。11(4) （1992)229 - 237。 [7] 大通崔 S.J.李,快速定位錯(cuò)誤的評(píng)估使用多維數(shù)組工件和機(jī)床觸摸探針,在:學(xué)報(bào)定位技術(shù)會(huì)議上,韓國(guó),2002 年,第 230 - 234 頁。 [8] 王瑞民 Lo,C.Y.蕭,刀位軌跡的方法補(bǔ)償重復(fù)加工過程,Int,j·馬赫。工具 Manuf。 38(3)(1998)205 - 213。 附錄二： Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system J.P. Choi a,? , B.K. Minb, S.J. Lee b a Graduate School of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea b Department of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea Abstract This paper suggests a method to reduce the machining errors of a three-axis machine tool by implementing an on-machine measurement with a touch probe. Probing errors of a touch probe and positioning errors of a machine tool, inevitably included in the measurement data, are compensated for to obtain the true machining errors for the repeated machining process. Positioning errors of a tool/probe tip are modelled by approximating error components as polynomial functions and considering the effects of backlash errors. To estimate the unknown model parameters, a cube array artifact composed of eight cubes is proposed and calibrated on a CMM. Simulation results and verification experiments showed that the measured and predicted positioning errors agree well within less than 10m for all axes. The actual cutting test of a simple block and two-dimensional curves showed that the machining errors are reduced to within 10m after error compensation. ? 2004 Published by Elsevier B.V. Keywords: On-machine measurement (OMM); Touch probe; Cube array artifact; Error compensation 1. Introduction In conventional manufacturing process, part inspection is done with stand-alone measurement instruments such as coordinate measuring machines (CMM), which are generally located at a separate room apart from a machine tool. This increases the overall manufacturing cost and time to obtain the final product, and the bottleneck phenomenon may be caused by the product stagnation due to the time lag between the machining and inspection process in case of the flexible manufacturing system. Furthermore, it is hard to transfer, fixture, and measure the complex, large-sized parts [1]. To overcome these problems, an on-machine measurement (OMM) system as illustrated in Fig. 1 is implemented using a commercial touch probe (MP10 from Renishaw Inc.). A touch probe is a relatively inexpensive an easy-to-use accessory that can deliver significant reductions in production time and cost and widely used for process improvement—automating and speeding part processing, even eliminating part errors of the process. The system is composed of optical module probe (OMP) and optical Corresponding author. E-mail addresses: feel2@korea.com (J.P. Choi), bkmin@yonsei.ac.kr (B.K. Min), sjlee@yonsei.ac.kr (S.J. Lee). machine interface (OMI). OMP, located between the probe head and the shank, receives machine control signals and transmits probe signals. Communication between the probe and the OMI is done via the optical transmission system, whereas RS-232 serial communication is used to transmit the measurement program (macro program) to the CNC controller and receive the measured data for further analysis using a personal computer. Fig. 2 shows the overall work flow of this research to enhance the machining accuracy by the on-machine measurement and error compensation system. NC data generated using the part model is fed to the CNC controller for use in the first-step machining. After the machining process is finished, the touch probe replaced with a cutting tool starts the measurement in the normal direction to the machined surface. Since a touch probe measures parts moving along the erroneous machine tool axes, the measured data inevitably include the probing errors originated from the structural characteristics of a touch probe, and the positioning errors originated from the inaccurate axis motion of a machine tool. These errors should be eliminated from the measured data to obtain the true machining error. If the true machining error is larger than the given tolerance, the new tool path is generated using the error compensation algorithm for the next-step machining. Machining and on-machine measurement processes are repeated until the required part tolerance 924-0136/$ – see front matter ? 2004 Published by Els evier B.V. J.P. Choi et al. / Journal of Materials Processing Technology 155–156 (2004) 2056–2064 2057 Fig. 1. On-machine measurement system. is obtained, resulting in the closed-loop machining system [2]. This paper suggests a methodology to quickly assess the positioning errors of a machine tool using a new error model and a cube array artifact. The error model is constructed by approximating error components appeared in the volumetric error model with polynomial functions. The error model is decomposed into forward and backward model according to the axis movement direction of a machine tool, because the backlash errors affect the on-machine measurement data. To determine the unknown model coefficients, a cube array artifact composed of eight cubes is proposed and calibrated on a CMM. Simulation results of the positioning errors at cube vertices showed that the estimated errors agree well with the measured errors for all axes in both forward and backward directions. A step gauge is used to verify the suggested error model. Finally, the machining tests of a simple block and two-dimensional curves are performed, where an error compensation method based on the line segmentation algorithm is applied to reduce the machining errors. It can be concluded that the machining errors are reduced to within 10m after error compensation. 2. Characterization of probing errors and positioning errors 2.1. Probing errors In touch probes, the mechanical structure supporting the stylus serves as the electrical switch that is triggered when the stylus is displaced. This results in probe lo bing with a three-lobed structure reflecting the triangular mechanical structure within the touch probe [3]. Since these probing errors affect differently the measurement data according to the probe approach direction, they must be compensated before performing the actual measurement. Fig. 3 shows the probing errors obtained by measuring a precise ring gauge with a diameter of 29.998 mm. The stylus length is 50mm and the probe ball radius is 1 mm. The magnitude of the probing errors is dependent on the stylus length and the orientation of the probe. After error compensation, the probing errors are reduced to within 5 m, which is the same order of the Repeata bility of a machine tool. 2.2. Mathematical formulation of machine tool errors Machine tool errors are propagated into the on-machine measurement data, since a touch probe measures parts moving along the erroneous machine tool axes. So, these errors Fig. 2. Workflow of on-machine measurement and error compensation system. J.P. Choi et al. / Journal of Materials Processing Technology 155–156 (2004) 2056–2064 Fig. 3. Compensation of probing errors should be identified and eliminated from the on-machine measurement data to obtain the true machining errors for the next-step machining process. To determine the positioning errors at any position within the work space, the general homogeneous transformation matrices (HTM) are used, which represent the coordinate transformation from the coordinate system of the rigid body frame to that of the reference coordinate system [4]. Multiplying the H T Ms for the moving elements and their error matrices successively from the reference coordinate system to the tool coordinate system actual positions are obtained in terms of ideal positions and all error components of a machine tool. Fig. 4 shows the coordinate system of a three-axis machine tool used in this research, and the resultant positioning errors are derived in the following equation: Here, δ ii (i =x, y, z) denotes the linear errors along the it ha xis, δ i j (i, j =x, y, z and i = j) the straightness errors in the it h axis direction when moving along the j t h axis, ε i j the angular errors around the it h axis when the slide moves along the j t h axis, S i j the squareness errors between the corresponding axes. And a i, bi, c i are the origin offsets from the (i? 1) t h coordinate system to the it h coordinate system, and L the ideal tool length along the z-axis (Table 1). To predict the positioning of a tool/probe tip within the work space using Eq. (1), the measurement data of 21 error components should be required. Laser inter ferometer system is widely used to measure those errors with high accuracy, but it requires long calibration time and cost [5]. To assess the positioning errors in a more quick and easy way. Table 1 Fig. 4. Coordinate system of a column-traverse vertical machining center. Origin offset values between neighboring coordinate systems (unit: mm) J.P. Choi et al. / Journal of Materials Processing Technology 155–156 (2004) 2056–2064 2059 errors are considered as constant irrespective of axis positions. Substituting the approximated error components into the volumetric error model, the parameterised error model is obtained in the form of matrix equation: EFWD = B p where EFWD is the 3 × 1 error vector, B the 3 × 15 scalar matrix, p the 15 × 1 coefficient vector of unknown parameters. The subscript of the error vector denotes that the error model is applicable when axes move in the forward direction, where all error components are set to zero at the corresponding axes’ origin. The model parameter vector p can be easily determined using the least square estimator. p = (BTB)?1BTEFWD Since a touch probe measures parts along the machine tool axes, the backlash errors of error components of a moving axis affect the measured data in addition to the positioning errors at given position, and therefore they must be included in the error model. From the preliminary experiment results using a laser system, backlash errors are assumed constant irrespective of axis positions [7]. Substituting approximated error components including the backlash terms into the previous volumetric error model and rewriting in a matrix form, the error model in the backward direction is derived as follows: EBWD = EFWD + Fh J.P. Choi et al. / Journal of Materials Processing Technology 155–156 (2004) 2056–2064 where EBWD is the 3 × 1 error vector in the backward direction, EFWD the 3 × 1 error vector in the forward direction (same as EFWD of Eq. (2)), F the 3 × 18 scalar matrix, the 18 × 1 coefficient vector to be determined whose components are the backlash errors of error components. Note that the backward errors are obtained by adding errors originated from the backlash errors to errors in the forward direction. The model parameter vector h can be estimated similarly as before: h = (FTF)?1FT{EBWD ? EFWD} (5) 3. Estimation of model parameters and simulation results 3.1. Cube array artifact To determine the model parameter vectors p and h ofEqs. (3) and (5), a cube array artifact consisting of eight cubes as shown in Fig. 5(a) is proposed, which enables to measure the positioning errors in both forward and backward direction [7]. The artifact is calibrated with a coordinate J.P. Choi et al. / Journal of Materials Processing Technology 155–156 (2004) 2056–2064 2061 measuring machine, and then is installed on the machine tool table for measurement with a touch probe as shown on the right side of Fig. 5(b). The differences between CMM data and OMM data at cube vertices are used to generate the error vectors EFWD and EBWD of both forward and backward error models, respectively. The error vectors and nominal positions of cube vertices in the machine coordinate system are used to determine the model parameter vectors. 3.2. Simulation With the estimated model parameters, the positioning errors at cube corners are predicted and compared with the measured errors. Figs. 6 and 7 compare the simulated positioning errors with the measured errors for both forward and backward directions of x-axis and y-axis, respectively. In the figures, the quadratic and cubic error models mean that the error components are approximated with the firstand second-order polynomial functions, respectively. Fig. 6. Simulated and measured positioning errors of x-axis. Fig. 7. Simulated and measured positioning errors of y-axis. It can be seen that the cubic error model predicts the errors more accurately than the quadratic model and the differences between the predicted and measured errors are less than 5m for all measurement points. Also, the positioning errors have relatively large differences even at the same measurement points according to the axis movement direction, i.e., forward and backward, validating the suggested error model considering the axis movement direction. The true machining errors for the repeated machining process can be estimated by eliminating the positioning errors of a machine tool from the measured data with high accuracy. 3.3. Model verification using a step gauge A step gauge with nominal block size of 10mm and pitch of 20mm as shown in the left side of Fig. 8(a) is used to verify the error model. It is mounted on the machine tool table and measured in both forward and backward directions Fig. 8. Model verification using a step gauge. Fig. 9. Part geometry used in machining experiments. Measured positioning errors at block surfaces are compensated for by the predicted positioning errors using the suggested error model. The total positioning errors are reduced to within 5 m after compensation, and the backlash errors differences between the positioning errors with respect to the measurement direction are estimthe regression lines. It can be concluded that the suggested error model can predict the positioning errors with acceptable accuracy and compensate for the measured data to obtain the true machining errors for the next-step machining. Machining experiment 4.1. Part geometry and error compensation scheme The on-machine measurement system is applied to the machining test of a simple block composed of square and diamond features and two-dimensional curves as shown in Fig. 10. Comparison of machining errors of a simple block. Fig. 9. After the first-step machining is finished, the cutting tool is replaced with a touch probe, which measures the machined surface at the equally spaced measurement points. The probing errors and posi