立式高速銑削加工中心縱向進給機構(gòu)設(shè)計5張CAD圖

立式高速銑削加工中心縱向進給機構(gòu)設(shè)計5張CAD圖,立式,高速,銑削,加工,中心,縱向,進給,機構(gòu),設(shè)計,CAD 附錄一： 減少加工誤差的三軸機床機械測量和誤差補償系統(tǒng) 機械工程研究生院,延世大學(xué),首爾,韓國 機械工程系,延世大學(xué),首爾,韓國 文摘： 提出一個方法來減少加工誤差的三軸機床通過實現(xiàn)一個機器測量觸摸探針。觸摸探針探測錯誤和機床的定位錯誤,不可避免地包含在測量數(shù)據(jù),彌補獲得真實的加工錯誤的重復(fù)加工過程。 定位錯誤的工具/探針尖被逼近誤差建模組件作為多項式函數(shù)和考慮抵制錯誤的影響。 來估計未知模型參數(shù)、多維數(shù)組工件組成的八個數(shù)據(jù)集提出了 CMM 和校準。仿真結(jié)果和驗證實驗表明,測量和預(yù)測定位錯誤同意不到 10 米之內(nèi)所有軸。 一個簡單的塊的實際切削試驗和二維曲線表明,加工錯誤減少到后 10 米 內(nèi) 的 錯 誤 補 償 。 2004 愛 思 唯 爾 出 版 的 帳 面 價 值 。 關(guān)鍵詞:機械測量(介質(zhì));碰探頭,立方體數(shù)組工件;誤差補償 1、介紹 在傳統(tǒng)制造工藝,檢驗部分完成了獨立的測量儀器,如坐標測量機(CMM), 通常位于一個單獨的房間除了機床。這就增加了整體生產(chǎn)成本和時間獲得最終產(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、夾具和測量復(fù)雜的大型零件[1]。 為了克服這些問題,一個在機器測量(石)系統(tǒng)見圖 1，是使用一個商業(yè)實現(xiàn)觸摸探針(從英國 MP10 Inc .)。 觸摸探針是相對便宜和易于使用的配件,可以實現(xiàn)顯著減少生產(chǎn)時間和成本,廣泛用于過程改進自動化和加速處理一部分,甚至消除一部分錯誤的過程。光模塊的系統(tǒng)由探針(OMP)和光學(xué)機接口(OMI)。OMP,位于探測頭和柄之間,收到機器控制信號和傳輸探頭信號。探測器和 OMI 之間的通信是通過光傳輸系統(tǒng),而使用 rs - 232 串行通信傳輸測量程序(宏程序)CNC 控制器和接收的測量數(shù)據(jù)進一步分析使用個人電腦。 圖 2 顯示了本研究的整體工作流程提高機械加工精度的測量和誤差補償系統(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量加工表面的法線方向。自從接觸探頭沿著錯誤的措施部分機床軸,測量的數(shù)據(jù)不可避免地包括探測錯誤源于觸摸探針的結(jié)構(gòu)特點,和定位錯誤源于不準確的軸運動的機床。這些錯誤應(yīng)該取消從測量數(shù)據(jù)來獲得真實的加工誤差。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償算法。加工和機械測量過程不斷重復(fù),直到所需的部分公差,導(dǎo)致閉環(huán)加工系統(tǒng)[2]。 提出一個方法來快速評估機床的定位錯誤使用一個新的數(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式函數(shù)。前后誤差模型分解為模型根據(jù)機床的軸的運動方向,因為反對錯誤影響機器測量數(shù)據(jù)。系數(shù)來確定未知的模型,一個多維數(shù)組工件組成的八個數(shù)據(jù)集提出了 CMM 和校準。在立方體頂點定位錯誤的仿真結(jié)果顯示,估計錯誤也同意所有軸的測量誤差在向前和向后的方向。計是用于驗證一步建議誤差模型。最后,一個簡單的塊和二維曲線的加 工測試執(zhí)行,在一個基于線分割算法的誤差補償方法應(yīng)用于減少加工錯誤。它可以得出的結(jié)論是,加工錯誤減少到誤差補償后 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、表征探測錯誤和定位錯誤 2.1 探測錯誤 在觸摸探針,機械結(jié)構(gòu)支持手寫筆作為電觸發(fā)開關(guān),當筆流離失所。這個結(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)的機械結(jié)構(gòu) [3]。因為這些探測錯誤影響到測量數(shù)據(jù)根據(jù)不同的調(diào)查方法方向,他們必須得到補償,然后再執(zhí)行實際的測量。圖3 顯示了通過測量獲得的探測錯誤一個精確的環(huán)規(guī)直徑為29.998 毫米。球針長度 50 mm 和探針半徑為 1 毫米。探測誤差的大小取決于針長度和方向的調(diào)查。誤差補償后,探測錯誤減少到 5 米之內(nèi),同一訂單的機床的可重復(fù)性。 2.2 機床誤差的數(shù)學(xué)公式 機床誤差傳播到機器測量數(shù)據(jù),自從接觸探頭沿著錯誤的措施部分機床軸。所以, 這些錯誤應(yīng)該在機器的識別和消除測量數(shù)據(jù)獲取項目下一步加工過程的真實加工錯誤。確定工作空間內(nèi)的任何位置的定位錯誤,一般齊次變換矩陣(HTM),這代表剛體坐標系統(tǒng)的坐標變換幀的參考坐標系統(tǒng)[4]。增加移動元素及其誤差矩陣的 htm 先后從參考坐標系到實際工具坐標系位置得到的理想位置和機床的所有錯誤組件。圖 4 顯示了坐標系的三軸機床用于 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)表示線性錯誤面前是沿著軸,δij(i,j = x,y,z 和我= j)第 i 個軸方向的直線度誤差沿著 jth 軸時,εij 角錯誤在第 i 個軸滑動沿著 jth 軸移 動,Sij 之間的垂直度誤差對應(yīng)的軸。和人工智能,bi,ci 原點偏移量從(我?1)屆第 i 個坐標系坐標系統(tǒng),和 L 的理想工具沿著 z 軸長度(表 1)。 表 1 預(yù)測工具的定位/探針針尖在工作區(qū)使用 Eq。(1),21 個錯誤組件的測量數(shù)據(jù)應(yīng)該是必需的。激光干涉儀系統(tǒng)被廣泛用于測量這些錯誤與精度高,但它需要長時間校準時間和成本[5]。評估定位錯誤更加快速和簡單的方式,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系統(tǒng)。確定模型參數(shù),可以簡單地使用一個觸摸探針和工件。獲得參數(shù)誤差模型、線性和角度錯誤假設(shè)作為第一和二階多項式函數(shù)[6]。直線度誤差推導(dǎo)通過集成角錯誤和方形錯誤被視為常數(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)。誤差向量的下標表示誤差模型是適用于軸方向的移動,所有錯誤組件被設(shè)置為 0 的相應(yīng)軸的原點。模型參數(shù)向量 p 可以很容易地使用最小二乘估計量決定的。 自從接觸探頭沿機床軸的措施部分,反對錯誤的錯誤的組件移動軸影響測量數(shù)據(jù)除了在給定位置定位錯誤,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)的初步實驗結(jié)果,反對錯誤假定常數(shù)無論軸位置[7]。代替近似錯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)。誤差向量的下標表示誤差模型是適用于軸方向的移動,所有錯誤組件被設(shè)置為 0 的相應(yīng)軸的原點。模型參數(shù)向量 p 可以很容易地使用最小二乘估計量決定的。 p = (BTB)?1BTEFWD (3) 2060 自從接觸探頭沿機床軸的措施部分,反對錯誤的錯誤的組件移動軸影響測量數(shù)據(jù)除了在給定位置定位錯誤,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)的初步實驗結(jié)果,反對錯誤假定常數(shù)無論軸位置[7]。代替近似錯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 的錯誤方向的向量(2)),F×18 標量矩陣 h 18×1 的系數(shù)向量確定的組件是錯誤的反應(yīng)錯誤組件。注意,通過添加錯誤源于落后的錯誤得到反彈錯誤方向的錯誤。模型參數(shù)向量 h 可以估計同樣像以前一樣: h = (FTF)?1FT{EBWD ? EFWD} (5) 3 模型參數(shù)估計和仿真結(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),八個立方體組成的立方體數(shù)組工件如圖 5 所示(一個)提出,使測量定位錯誤都向前和向后的方向[7]。工件與坐標校準。 圖 5。多維數(shù)組工件和機器測量。 《材料處理技術(shù)雜志》155 - 156(2004)2056 - 2004 2061 2061 測量機,然后安裝在機床上的桌子與觸摸探針測量右側(cè)的圖 5 所示(b)。CMM 的區(qū)別在立方體頂點數(shù)據(jù)和介質(zhì)數(shù)據(jù)用于生成的錯誤矢量 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的位置在機器坐標系是用來確定模型參數(shù)向量。 3.2 模擬 估計模型參數(shù),定位錯誤在立方體角落預(yù)測,并與實測的錯誤。無花果。6 和 7 比較了模擬定位誤差與測量誤差都向前和向后 x 坐標軸和 y 坐標軸的方向,分別。數(shù)據(jù)的二次和三次錯誤模型意味著錯誤組件與 fistand 近似二階多項式函數(shù),分別。 圖 6。模擬和測量軸的定位錯誤。 圖 7。模擬和測量定位錯誤的 y 軸。 3.3 使用步驟計模型驗證 一步計10 毫米的名義塊大小和間距20 毫米的左邊圖8 所示(一個)是用于驗證誤差模型。它是安裝在機床表和測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 使用步驟計模型驗證 圖 9 幾何部分用于加工實驗 4 加工實驗 4.1 部分幾何和誤差補償方案 機器測量系統(tǒng)應(yīng)用于加工測試一個簡單的塊組成的廣場和鉆石的特性和二維曲線如圖 10 所示。 圖 10 一個簡單的塊比較加工錯誤 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,刀具被替換為一個觸摸探針,用于衡量的機械加工面等距的計量點。計量點的探測誤差和定位誤差從測量數(shù)據(jù)得到真正消除加工誤差考慮 探測器接近角。注意,調(diào)查的方法是不斷沿著角兩邊的廣場和鉆石的特性,而測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償點測量的點。補償點是由添加真實加工反對派方向[8]中的錯誤。第二次加工后使用 圖 11 比較二維曲線的加工錯誤 真正的加工錯誤。如果真正的加工誤差小于公差,這個過程完成后,得到最終的產(chǎn)品。否則,部分再次測量和加工誤差補償精度得到迭代,直到所需的部分。 4.2 加工結(jié)果 圖 10 顯示了加工誤差測量與加工第一步后觸摸探針第二次加工和測量完成后, 一部分是在 CMM 測量在同一測量與機器測量數(shù)據(jù)點比較。 5 結(jié)論 提出一個在機器測量和誤差補償系統(tǒng),減少使用觸摸探針加工錯誤。真實加工錯誤決心通過消除接觸探頭的探測錯誤和機床的定位錯誤。定位錯誤考慮抵制錯誤的影響嗎錯誤的組件。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)測定位錯誤同意所有軸的測量錯誤該機器測量系統(tǒng)可以擴展傳統(tǒng)機床的測量機精度檢驗和改進部分。 參考文獻： [1] 薩達姆政權(quán)金、D.H. Kim j .侯爾的機器上測量系統(tǒng)。Soc。摘要。Eng。18(6)(2001) 這 9 到 18。 [2] 薩達姆政權(quán)書釘、孫 J.W.林亭汝榮格,閉環(huán)方法減少總加工誤差:實驗和分析,反式。NAMRI /中小企業(yè) 15(1997)311 - 316。 [3]J.A.博世,坐標測量機和系統(tǒng),馬塞爾·德克爾公司,紐約,1995 年。 [4]交流可以用 Y.M. Ertekin,機床誤差模型的推導(dǎo)和誤差補償過程三軸垂直的加工2064 《料處理技術(shù)雜志》155 - 156(2004)2056 - 2064 中心使用剛體運動學(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)控加工工具的準確性在機器檢查,j . Manuf 系統(tǒng)。11(4) （1992)229 - 237。 [7] 大通崔 S.J.李,快速定位錯誤的評估使用多維數(shù)組工件和機床觸摸探針,在:學(xué)報定位技術(shù)會議上,韓國,2002 年,第 230 - 234 頁。 [8] 王瑞民 Lo,C.Y.蕭,刀位軌跡的方法補償重復(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