1360029 九、發明說明： 【發明所屬之技術領域】 本發明涉及一種機上檢測設備與量測方法，尤其是一種 機上二維度輪廓檢測的方法與系統。 【先前技術】 夕軸工具機一直是生產製造設備的主流，面臨現今國際 =般態的改變，國内工具機也持續地進行升級，朝向微型化、 鬲精密、高速化方向發展且有相當不錯成果。相關的研究開發 項目包含有精微工具機、高速線性馬達傳動式工具機、平行式 結構式工具機、複合式五軸加工機、高精密高速球、微加工 技術、高速切削技術、乾切削技術、精密量測技術、誤差補償 技，’皿升熱變形補償技術、高速進給控制技術、結構分析技 術等’主要朗應高科技產品如精賴具、半導體設備零件、 航太零件、光電產品之微小倾精密的需求，因此未來工具機 的性能指標勢必將朝向次微米、高精度方向來發展。 機上檢測設備與量測方法為重要的關鍵技術之一，適用 於進行高精度微虹件_度制設備，高_如掃描式電子 顯藏、電子微探分韻、聚鱗子鱗，巾健的如閉合回 ，光學系統……等’這些檢設備有一項共通點，即 為非接觸式制方式。對機虹件的加I精度制技術，現 今相關技術及城大娜職_量财式，適用於精微 =具===、機地啦軸上誤祕正的相關研 究’也為刻不谷緩的重點。 6 【發明内容】 解決上述傳統之機上 之標的 求，本發明提 1之t明背景中，為了符合產業上某些利益之需 ㈣上1^一種機上二維度輪廓檢測的方法與系統可用以 維度輪廓檢測的方法與系統未能達成 與系、ί發明之—目的係提供—種機上二維度輪雜測的方法 法，實、1祕—職上二較麵檢測的方 ，幟台之-刀具依據一數值控制碼於一 1件上姻： ，亡^驟⑽與步驟咖所示，以一圖形擷轉置操取工件 一影像’並且以—邊緣_法依據影像產生影像巾的一工件 S。另外’如步驟1340所示，依據數值控制碼產生一理論 ^廓。接下來’如步驟測所示，將工件輪廓與理論輪靡疊 口於同一座標系。之後，如步驟1360與1370所示定義理論 ，廓在座標系上的複數個理論切削點，並且定義工件輪廓在: 標系上的複數個真實切削點。再接下來，如步驟1380所示， 依據每一個理論切削點分別找出與每一個理論切削點相匹配 的真實切削點。最後，如步驟1390所示，依據每一個理論切 削點與相匹配的真實切削點分別識別出相應於每一個理論切 削點的一加工誤差。 本發明之另一具體實施例係一種一種機上二維度輪廓檢 测的系統，包含一工件、一内儲一數值控制碼的一電腦數值控 制機台與一誤差辯識裝置。其中電腦數值控制機台依據一數值 控制碼以-刀具於-工件上切削 以是一種_式碼之電腦，内儲之程式碼置^ 之裝，’包含.w-_操取裝置操取工件之_影像穿: 以-邊緣_法依據影像產生影像中的—置’ 據數卿恤，_购；糾後 定義理論輪靡在座標系上的複:； 理响切削關裝置，定紅件麵在座射、 m驢；依據每-個理論切難分顺出與每—個理2 削點相真實靖點㈣置;傾每—伽 2 匹配的真實切舰分別朗出相應於每-個理論切^的一 加工誤差的裝置。 月U的 【實施方式】 方法此所探討的方向為一種機上二維度輪廓檢測的 為了能徹底地瞭解本發明，將在下列的描述中提 出评盡的步驟及其組成。顯然地，本發明的施行並未限定 亡-維度輪雜測的方法與系統之技藝者所熟習的特殊細 即。另-方面’眾所周知的組成或步驟並未描述於細節中以 避免造成本發明不必要之關。本發明輸佳實施例合詳細^ 述H，然而除了這些言羊細描述之外，本發明還可以“ =範:=例中’且本發明的範圍不受限定，其以之後的 一本發明提出一種機上二維度輪廓檢測法，如第一圖所 不。首先，如步驟110所示，先將工件切削後，利用相機直接 機上榻取卫件影像。之後，如步驟12G所示，以邊緣_法 5析工件麵，在本發明之—較絲财，邊緣伽彳法是採用 Canny邊緣伯测法。同時，如步肆13〇與步驟14〇所示以护 驗片刀析工件影像的像素解析度，並且重建以像素為單二 =Ϊ工件輪廊。接下來，如步驟150所示，將理論輪靡‘ 實維度區域進行座標系定義與鋪轉換(亦即理論與 牙、牛輪靡座標系定義與輪廓重疊貼合）。再接下來，如+ 差辨識!計算誤差，在本發明之-較佳範i 德备〃為早絲#算誤差。最後，如步驟170所示，依 像素解析度來進行誤差長度單位轉換。 件〜機加讀虹倾，械在機上進行工 牛娜’械與工叙咖方位義如第二_示 f 〇：D H件〇進行影像棘。〇 知白攝時相機座標系“ z/，1]τ與工件座標 的ζ軸向為平行。 yei 廓，在tir攝後’利用邊緣辨識分析影像中的實際工件輪 F工产較佳範例情用以卿邊緣偵測法來分析實 此為加工後真實工件輪靡。在分析實際工件二 與加工刀具，她合刀具切削路經 求得加工縣，在ίί接進行輪叙财異比對進而 論工件輪廓上八別^ ^較錄财在實際轉輪廓與理 係來定義實際件輪由此兩參考點之間座標關 區或座軚系，並經由座標轉換後使 1360029 兩輪廓在同一座標系上（理論工件輪廓區域座標系）。當兩輪 廓經座標轉換有相同座標系後，利用誤差原理配合影像匹配方 式發展的誤差辨識法來計算影像中工件的加工誤差。最後，再 根據影像解析度將誤差由像素單位值轉換為長度單位值，此時 單位轉換後的誤差即為微型工件在工作空間中的輪廓誤差及 循跡誤差。 機上二維度輪廓檢測法步驟主要包含四項：（1).邊緣辨 識。（2).以像素單位之理論工件輪廓重建。（3).定義實際與理 論工件輪廓之區域座標系及二輪廓間座標轉換。（4).誤差 識。 邊緣辨識 邊緣是指影像中局部灰階變化最顯著的部分，邊緣主要 存在於目標與目標、目標與背景、區域與區域（包括*同色彩) 間，疋影像分割、紋理特徵和形狀特徵等影像分析的重要基 礎而邊緣檢測即是利用對鄰域灰階值的偏微分來計算像素的 梯度後’舰合梯度Η檻值的設絲_像敍㈣邊緣。邊 緣檢測可分析影像巾物件大小、綠#，料，亦可從影像中 抽取特定物體，以及辨識或分類物體。邊緣檢測—般分為 四項步驟： 〇)影像濾波··由於現今影像擷取模式大都採用數位式， CCD CMOS等。而數位式的感光元件因内在電流、曝光 時間長短、感紐的設定、光_大小等㈣響，在影像中會 標 〜Γΐϊϊ?化:利用一階或二階偏微分來進行梯度的計 异，' $又來代表像素其鄰域的灰階變化，稱為邊緣強化。 -(3)邊，分析：經由邊緣強化後’影像皆以梯度來進行標 不為了綠定那些像素是邊緣，可利用梯度配合邊緣分析來判 別像素使否為邊緣。最簡單的邊緣分析為梯度門艦值 (Threshold)判別法，亦可稱為梯度閥值；當像素的梯度超過 所設的門檻值時判別為邊緣，反之則否。 (4)邊緣定位：如果經_後確定像素為邊緣後，則以二 值影像來標記其在像素座標系上的座標。 索貝爾、高斯拉普輯及卡尼邊緣伽彳法為常見的邊緣 檢測=式。索貝爾及高斯拉普拉斯邊緣偵測法主要利用偏微分 濾波器來計算像素的梯度，並根據梯度門插值的設定來判別是 否為邊緣，雖然步驟非常的簡單及快速，但是單由梯度來判別 邊緣，會造成邊緣檢繼影像上出❹重、不正確及不連續的 邊緣，此狀況將不利於標示工件輪廓。卡尼邊緣偵測法結合了 口高斯(Gaussian)遽波器的遽波功能及索貝爾(s〇bd)偏微^波 器其具有方向性的優點，特點在於能從影像上來標示單一且最 佳的邊緣，並配合雙閥值判別法對不連續的邊緣進行連接。圖 第三A圖為五角大廈空關，經由索貝爾、高斯拉普拉斯及 卡尼邊緣偵測後的邊緣影像，如圖第三B圖、第三圖c及圖 第三；〇_示。由第三A® '第三b圖、第三圖c及圖第三 D圖令可明顯比較得知’卡尼邊緣偵測法找出的邊緣在影像上 =示較稍，且邊緣㈣單—像素來絲 較佳範例中為了找出最佳、在本U之- 緣_法由影像中檢測工件的麵牛輪廓，採用卡尼邊 像素單位之理論碍輪廓重建1360029 IX. Description of the Invention: [Technical Field] The present invention relates to an on-board detecting device and measuring method, and more particularly to a method and system for two-dimensional contour detection on a machine. [Prior Art] The Xi axis machine tool has always been the mainstream of manufacturing equipment. Facing the current international change, the domestic machine tools are also continuously upgraded, and the development is toward miniaturization, precision, and high speed. Results. Related research and development projects include micro-tooling machines, high-speed linear motor-driven machine tools, parallel-structured machine tools, composite five-axis machines, high-precision high-speed balls, micromachining technology, high-speed cutting technology, dry cutting technology, Precision measurement technology, error compensation technology, 'dish heat deformation compensation technology, high-speed feed control technology, structural analysis technology, etc.' mainly should be high-tech products such as precision rigs, semiconductor equipment parts, aerospace parts, optoelectronic products The demand for precision is small, so the performance indicators of future machine tools are bound to develop in the direction of sub-micron and high precision. On-board inspection equipment and measurement method is one of the important key technologies, suitable for high-precision micro-red parts _ degree system, high _ such as scanning electronic display, electronic micro-exploration, multi-scale scales, towel The health of the closed circuit, the optical system, etc. 'These inspection devices have a common point, that is, the non-contact system. Adding I precision technology to the machine's rainbow parts, today's related technology and the city's major _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Slow focus. 6 [Summary of the Invention] To solve the above-mentioned conventional on-board target, the present invention provides a method and system for inspecting two-dimensional contour detection on the machine in order to meet certain interests of the industry. The method of dimension contour detection and the system failed to reach the system, and the purpose of the invention is to provide the method of two-dimensional round-wheel miscellaneous measurement on the machine, and the first and the second secrets. - The tool is based on a numerical control code on a piece of marriage: , the death of the step (10) and the step coffee, as shown in the figure, the workpiece is imaged by a graphic transposition and the image is produced by the edge-based method. A workpiece S. In addition, as shown in step 1340, a theoretical profile is generated based on the numerical control code. Next, as shown in the step measurement, the contour of the workpiece is overlapped with the theoretical rim in the same coordinate system. Thereafter, as defined by steps 1360 and 1370, the theory is defined by a plurality of theoretical cut points on the coordinate system, and the workpiece contour is defined as: a plurality of true cutting points on the scale. Next, as shown in step 1380, a true cutting point that matches each of the theoretical cutting points is found for each theoretical cutting point. Finally, as shown in step 1390, a machining error corresponding to each of the theoretical cutting points is identified based on each of the theoretical cutting points and the matching true cutting points. Another embodiment of the present invention is a system for two-dimensional contour detection on-board, comprising a workpiece, a computer numerical control machine for storing a numerical control code, and an error identification device. The computer numerical control machine is based on a numerical control code to cut the tool on the workpiece to be a type of computer, the stored code is set to ^, and the .w-_ operation device is used to take the workpiece. _Image wear: By - edge _ method according to the image generated image - set ' _ number of qing, _ purchase; after the definition of the theory of the rim on the coordinate system:; The surface is shot, m驴; according to each theory, it is difficult to divide and divide with each of the 2 points of the point of the true point (4); the actual cut of each of the - gamma 2 matches is corresponding to each A device for processing a machining error. [Embodiment] The method discussed herein is an on-board two-dimensional contour detection. In order to thoroughly understand the present invention, the steps and composition thereof will be described in the following description. Obviously, the practice of the present invention does not limit the particular details familiar to the methods of the death-dimension wheel mismatch and the skill of the system. The other aspects are not described in detail to avoid obscuring the invention. The preferred embodiment of the present invention will be described in detail with reference to H. However, in addition to these detailed descriptions, the present invention may also be "in the case of:" and the scope of the present invention is not limited, and the following invention is An on-board two-dimensional contour detection method is proposed, as shown in the first figure. First, as shown in step 110, after the workpiece is cut, the camera is used to directly access the image of the workpiece. Thereafter, as shown in step 12G, The surface of the workpiece is analyzed by the edge method. In the present invention, the edge gamma method is performed by the Canny edge method. At the same time, the workpiece is analyzed by the blade as shown in step 〇13〇 and step 14〇. The pixel resolution of the image is reconstructed with pixels as the single two = Ϊ workpiece wheel gallery. Next, as shown in step 150, the theoretical rim 'real dimension area is coordinated and tiled (ie, theory and teeth, The definition of the cow rim coordinate system overlaps with the contour.) Next, as the + difference is recognized! The calculation error is calculated in the present invention - the preferred error is the error. Finally, as shown in step 170 The error length unit conversion is performed according to the pixel resolution. ~ Machine plus reading the rainbow tilt, the machine on the machine to carry out the work Niu Na 'mechanical and industrial Syrian coffee position as the second _ show f 〇: DH pieces 〇 carry out image thorns. 〇 know the white camera coordinates coordinate system "z /, 1] τ is parallel to the ζ axis of the workpiece coordinate. The yei profile, after tir shooting, uses edge identification to analyze the actual workpiece wheel in the image. F is a good example of the use of the edge detection method to analyze the real workpiece rim after machining. In the analysis of the actual workpiece 2 and the machining tool, she combined the cutting path of the tool to obtain the processing county, and then carried out the round-to-counter comparison in the ίί, and then on the contour of the workpiece, eight pieces of ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Define the actual part wheel between the two reference points, the coordinate area or the coordinate system, and convert the coordinates to make the 1360029 two contours on the same coordinate system (the theoretical workpiece contour area coordinate system). When the two-wheel profile is converted to the same coordinate system, the error principle is used to calculate the machining error of the workpiece in the image by using the error identification method developed by the image matching method. Finally, according to the image resolution, the error is converted from the pixel unit value to the length unit value. At this time, the error after unit conversion is the contour error and tracking error of the micro workpiece in the workspace. The on-board two-dimensional contour detection method mainly includes four items: (1). Edge recognition. (2). Theoretical workpiece contour reconstruction in pixel units. (3). Define the coordinate system between the actual coordinate system and the contour of the workpiece. (4). Error knowledge. The edge recognition edge refers to the most significant part of the local gray scale change in the image. The edge mainly exists between the target and the target, the target and the background, the region and the region (including * the same color), and the image segmentation, texture features and shape features. The important basis of the analysis and edge detection is to use the partial differential of the gray-scale value of the neighborhood to calculate the gradient of the pixel after the 'shipping gradient Η槛 value of the silk_image (4) edge. Edge detection analyzes image object size, green #, material, and extracts specific objects from the image, as well as identifying or classifying objects. Edge detection is generally divided into four steps: 〇) Image filtering · Because most of today's image capture modes use digital, CCD CMOS and so on. The digital type of photosensitive element due to internal current, exposure time, sensitivity setting, light_size, etc. (four) ring, in the image will be marked ~ Γΐϊϊ 化: using first or second order partial differential to carry out the gradient of the difference, ' $ again represents the grayscale change of the neighborhood of the pixel, called edge enhancement. - (3) Edge, Analysis: After edge enhancement, the images are all graded with gradients. For those pixels that are green, the edges are edged. Gradient and edge analysis can be used to determine whether the pixel is an edge. The simplest edge analysis is the gradient gate threshold (Threshold) discriminant method, which can also be called the gradient threshold; when the gradient of the pixel exceeds the set threshold value, it is judged as the edge, otherwise it is not. (4) Edge positioning: If the pixel is determined to be an edge after _, the coordinate on the pixel coordinate system is marked with a binary image. The Sobel, Gaussian, and Kani Marginal methods are common edge detections. The Sobel and Gaussian Laplacian edge detection method mainly uses a partial differential filter to calculate the gradient of the pixel, and determines whether it is an edge according to the setting of the gradient gate interpolation. Although the steps are very simple and fast, the gradient is only Discriminating the edges will result in heavy, incorrect, and discontinuous edges on the edge detection image, which would be detrimental to marking the contour of the workpiece. The Kani edge detection method combines the chopping function of the Gaussian chopper and the directionality of the Sobel (s〇bd) micro-wave device, which is characterized by the ability to mark the single and the most from the image. Good edge and double-threshold discriminant method for connecting discontinuous edges. Figure 3A shows the edge image of the Pentagon, which is detected by Sobel, Gauss Laplace and Kani Edge, as shown in Figure 3B, Figure 3C and Figure 3; . From the third A® 'the third b, the third c and the third D, you can clearly see that the edge found by the Kani edge detection method is on the image = the edge is slightly smaller, and the edge (four) is - In the preferred example of pixel-on-wire, in order to find the best, in the U-edge method, the face contour of the workpiece is detected by the image, and the theoretical contour reconstruction of the pixel unit of the Kani side is used.
法得論工件麵觀與實際岐細取方 J之實際工件輪辄較求得，然而在CAD 影像方法擷取之實際工件輪廟的單位不 以像素Λ健’喊雜職咐際工件輪靡 據。因此，料·，，、/纟直制絲較騎讀誤差的依 能進5 ^料㈣理論工件輪廊模型，方 χγ 工# 77析’在本發明之一較佳範例中利用1^碼中 Z用m補路购析配合刀具尺寸及影像解析度的搭 认工件於产看^像素為单位的理論工件輪廓。像素單位之理 :工件輪廓重建流程如第六圖所示，理論工件麵重建步驟如 卜· (1).分析工件影像拍攝時的影像解析 t每單位像素代表的真實物件尺寸），如步驟= ;^驗片（权驗片如第四圖所示）分析拍攝時的影像解析度，步 驟為： (a) .校驗片置於工件的表面。 (b) .利用相機拍攝校驗片的影像。- (c) .利用邊緣檢測法分析校驗片上圖像的輪廓。 像上上給㈣像喊實財（_)與影 解折声/郛尺寸㈣叫進行換算後，即可求得影像 解析度（mm/pixel)。 並由62G所不’從Nc碼掏取平面刀具切削路徑， 徑崎度，糊單轉換棟砰㈣具切削路 又雜值無為像素單健，最後在參考座上以二 影像來標記平面刀具切削路徑的座標。/ ’、 刀且m步Γ3ϋ與娜64G啦，儀雜_法，對 :::路麵職求得詳細刀具切削路徑。CAD軟體在 :』====通常:對線段兩端點的刀 切除的£域，以分析其理論 中利用線性内插法求得在本發明之一較佳範例 標。以内ί直線具補路徑上各路徑像素的座 量在ζ.虚直?刀具切削路捏的座標時，Γ以一個像素增 向有進行’座標的計算’亦即在與ζ·2之間在’轴 模嘱"，若伽咖情況則以四捨五入 (4).如步驟650盥步驟心ςη沉· - , 度將刀具半徑_換為值·求制影像解析 路徑所示，利用單位轉換後的詳細刀具切削 平面的上Γ在參考座標系上重建理論加工模型。建立 爽八狀+ r型時，_詳細刀具_路徑配合刀具半徑 來刀析靖轉上翁_麵，結4雜上標記& 1360029 標，最後分析完成後的二值影像即為理論加工模型。 (6).如步驟680所示，利用邊界追蹤原理（Boundary Following Algorithm)搜尋理論加工模型的邊界，分析後的邊 界即為理論工件輪靡。 第五A圖與第五B圖為二值影像之邊界追蹤的範例，第 五A调為原始的二值影像’第五B圖為邊界追蹤後的邊界圖。 定義實際工件輪廓與理論工件輪廓之區域座標系及二輪廓間座標 轉換 實際工件輪廓與理論工件輪廓在求得後，理論上實際工 件輪廊可經由座標轉換方式將其轉換至理論工件輪廓座標系 上以進行誤對，但實際轉麵上之參考座鮮、與理論輪 ，上之像素座標系之間的轉換參數需有複雜的檢測方式求 仔。有鑑於此’在本發明之一較佳範例中將定義實際工件輪廊 與理論工件輪賴域麵系，並藉由祕域座標系之間的座標 ，換矩陣’將實際I件輪廓貼合於理論工件輪廓上，以便進行 、描辨識第七圖為疋義實際工件輪廓與理論工件輪廓之區域 上7^系及二輪廓間座標轉換流程，步驟如下： 舢η立如步驟702所示，訂定加工參考座標系原點及其X 兩參考點。再如步驟71G所示，定義理論工件輪 ^座Μ (4,人）與工件輪廓區域座標系（We,)，以作為 祕祕準。錢立CAD__其參 廓^奸ί點及工件上的任意—賴為基準，以定義工件輪 齡或座杨，該二點祕社件加㈣魅於工件上，屆時 1360029 可用來疋義實際工件上作貼合用的工件區域座標系。隨後，如 步驟712所示’設置理論工件輪廓座標系與輪廓於像素座標系 平面上並求其的位置與方位。 (2).如步驟704所示，藉由工具機加工，以加工參考座 標系原點上加工的參考點及另加工參考點作為兩參考點。再如 步驟72G所示，建立實際功輪親域座標系n)與理論 工件輪廓區域座標系（心乂）之間的座標轉換矩陣。實際工件 輪廓區域座標系與理論1件輪躯域座標系之間的轉換 可表示為： 、 ΛThe method of the workpiece is compared with the actual workpiece rim of the actual 岐 取 取 J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J J according to. Therefore, the materials,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, In the middle of Z, the m is used to purchase and analyze the contour of the tool and the resolution of the image. Pixel unit principle: The workpiece contour reconstruction process is shown in the sixth figure. The theoretical workpiece surface reconstruction step is as follows: (1). Analyze the image resolution of the workpiece image when t is represented by the real object size per unit pixel), as in step = ; ^ The test piece (as shown in the fourth figure) analyzes the image resolution at the time of shooting, the steps are: (a) The check piece is placed on the surface of the workpiece. (b). Use the camera to take an image of the patch. - (c) . Use edge detection to analyze the outline of the image on the patch. The image resolution (mm/pixel) can be obtained by converting the image to the top (4) and the real image (_) and the image. And the 62G does not 'take the plane cutting path from the Nc code, the diameter is the roughness, the paste conversion is the dong (4) with the cutting path and the miscellaneous value is the pixel single health, and finally the plane tool cutting is marked by the two images on the reference seat. The coordinates of the path. / ‘, knife and m step Γ 3 ϋ and Na 64G, Yi Zao _ law, on ::: pavement job to obtain a detailed tool cutting path. The CAD software is: 』==== Normally: the field of the knife cut off at the ends of the line segment is analyzed in the theory to obtain a preferred example of the present invention by linear interpolation. Within the ί line, the amount of pixels of each path on the path is in the ζ. virtual straight? The coordinates of the cutting path of the tool cutting, Γ is increased by one pixel and there is a 'coordinate calculation', that is, between ζ·2 'Axis mode 嘱", if the gamma case is rounded off (4). If step 650 ς step ς · sink · - , the degree of the tool radius _ is changed to the value of the image analysis path, using unit conversion The upper jaw of the detailed tool cutting plane reconstructs the theoretical machining model on the reference coordinate system. When creating a cool eight-shaped + r-type, the _ detailed tool _ path with the tool radius is used to analyze the jing _ _ _ surface, the knot 4 miscellaneous mark & 1360029 standard, the final analysis of the binary image is the theoretical processing model . (6). As shown in step 680, the boundary of the theoretical machining model is searched by the Boundary Following Algorithm, and the analyzed boundary is the theoretical workpiece rim. The fifth and fifth B are examples of boundary tracking of binary images, and the fifth A is the original binary image. The fifth B is the boundary map after the boundary tracking. Defining the actual coordinate of the workpiece and the contour of the theoretical workpiece and the coordinate between the two contours. After the actual workpiece contour and the theoretical workpiece contour are obtained, the actual workpiece wheel gallery can be converted to the theoretical workpiece contour coordinate system by coordinate conversion. In order to make a mistake, the conversion parameters between the reference set on the actual transfer surface and the theoretical wheel and the pixel coordinate system on the upper surface need to be complicated. In view of the above, in one preferred embodiment of the present invention, the actual workpiece wheel gallery and the theoretical workpiece wheel lining system will be defined, and by the coordinates between the coordinate system of the secret domain, the matrix will be replaced by the actual profile. On the theoretical workpiece contour, in order to carry out and trace the seventh figure as the coordinate conversion process between the actual workpiece contour and the theoretical workpiece contour region, the steps are as follows: 舢η as shown in step 702, Set the machining reference coordinate system origin and its X two reference points. Further, as shown in step 71G, the theoretical workpiece wheel Μ (4, person) and the workpiece contour area coordinate system (We,) are defined as secrets. Qian Li CAD__ its profile ^ ί ί and any part of the workpiece - as a benchmark to define the workpiece wheel age or seat Yang, the two points of the secret member plus (four) charm on the workpiece, then 1360029 can be used to deny the actual The workpiece area coordinate system used for bonding on the workpiece. Subsequently, as shown in step 712, the theoretical workpiece contour coordinate system is set and contoured on the pixel coordinate system plane and its position and orientation are sought. (2) As shown in step 704, by machine tooling, the reference point processed on the origin of the reference coordinate system and the other processed reference point are used as two reference points. Further, as shown in step 72G, a coordinate transformation matrix between the actual power wheel home coordinate system n) and the theoretical workpiece contour region coordinate system (heart palpitations) is established. The actual workpiece contour area coordinate system and the theoretical one-wheel body coordinate system can be expressed as: , Λ
Cos0 -SineJjc, βΐηθ Cose i. (l) 其中 h -座標系間在ζ·及y方向的平移距離 Θ:二座標系間的旋轉角度 卩遺後’如步驟722所示，於拍照後求得實際工件輪廊座 不、於像素座標系位置與方位。因此，如步驟73〇所示，經由 ^矩陣轉後，可轉實際王件輪顧域座標倾理論 輪廓座標系之間的座標轉換式。 糾你⑽如步驟740所示，利用座標轉換式將實際工件輪鄭 工:Γ碰似伟，並_ ^檢職鱗分析新的實際 β轉二° ^下來’如步驟75G所示’由於實際卫件輪靡經座 =換置可能不落於影像網格，因此需利用工件影 仃4轉換並經由邊緣檢測來重新定義實際工件輪廓。 1360029 貼合= 輪=上將 誤差辨識法 誤差編輪輪。⑵.猶跡Cos0 -SineJjc, βΐηθ Cose i. (l) where the translation distance between the h-coordinates in the ζ· and y directions Θ: the rotation angle between the two coordinate systems 卩 after the 'step 722, after the photo is taken The actual workpiece wheel gallery is not in position and orientation of the pixel coordinate system. Therefore, as shown in step 73, after the matrix is rotated, the coordinate conversion between the theoretical coordinate coordinate systems of the theoretical coordinate system can be transferred to the actual king. Correcting you (10) As shown in step 740, using the coordinate conversion formula to actualize the workpiece wheel: bumpy like Wei, and _ ^ check the scale analysis of the new actual β turn 2 ° ^ down 'as shown in step 75G' due to the actual Guard rims = The replacement may not fall on the image grid, so the workpiece 仃 4 transformation is used and the actual workpiece contour is redefined via edge detection. 1360029 Fit = Wheel = Admiral Error Identification Method Error wheel. (2).
服匹配不撕狀，_導致物_路徑偏_先規U 切削時刀具所在實際位置Pa與理論路徑之 =的最短麟，ε。轉跡誤㈣所指岐在 實際刀具位置Pa和理論位置Pd之間的距離。 部 採加誤差與循跡誤差時需已知刀具實際位置，因 Γ朗I件的方式此法可_彳真實的加工誤 差’…、、而缺點疋並無法由實際工件輪廊 二種誤差的㈣，需在已知理論 具鲫時實際加工位置（Pa)和對應的 差及追蹤誤差，因此在本發明之 中最重識方法，以因應上述問題。在誤差辨識方法 刀為如何以輪廓匹配方式找出刀具實際加工位 _ehing ^ 例方法中將以微小匹配視窗 理掄糾# ⑺城衫式，比對實際功局部輪廓與 似性’進而決定對應理論刀具位置的實際刀 假鄕 _於微型工具機已具-^可靠精度，藉由下兩項 觀概念細 =======差，以微 由輪廓分佈方式情況，二設 =:= 1360029 麵。(2).在刀具切 内。二真實刀具切削點將落在理論刀具切削點的誤差半徑 利用旦基Γ上述的兩項假設’當各理論域點座標求得後，可 像轉財__棘。在進行影 ί = 須情其搜尋條件及搜尋目標進行定義：⑴ 二切肖彳额及理論輪齡齡論糊點，當刀具切削 /、理論輪廓之間有最短距離時，定義此理技 ^ 工 點；（2)將各理論切_愈㈣从’ ”郛‘』為理确切削 认工躲〜與細工件輪標設為影像A，理 二義為影^的特徵點。⑶將實際工件輪靡 Μ、Γ為象實際玉件輪廓座標定義為影像B的特徵點。 理論切削點設為目標視窗中心的座標。⑺依工具機的 圓半徑，此鄰近圓半徑依影像解析度轉換為 像=早位後，作為影像之特徵匹配原理中的搜尋視窗搜尋範圍 1 Candocia 的相似性參數公式作為切削輪廓匹配相似性計算之用。 當各理論_點紐尋最麵對應真實娜點後， 根據輪廓誤差與循跡誤差在空間上的㈣，分析加卫後的 誤差與循跡誤差。以第十圖為例，ρι為刀具切削路徑〇時的 理論切削點’而P2為實際切削點。根據輪廓誤差的原理在 理論工件輪紅找到P3 ’作為與p2之間存在最短距離，P2 到P3之間的距離即表示刀具切削路徑C1時工件的輪廊誤 差，E卜根據循跡誤差的原理，P2到ρι之間的距離即表示刀 具切削路徑C1時的循跡誤差，E2。 衫像之特徵匹配原理 影像匹配主要為搜尋影像空财物件關職關係對 ^糸的疋義如下.影像平面中的物件點，必射在對應的影 罔面中搜尋到其對應點。第十—圖為影像之特徵匹配原理流 私圖’步驟如下： 如步驟1112、步驟1114與步驟1116所示，將理論切 削位置设為目標視窗中心，而視窗所涵蓋的理論工件 ，，目標視窗(如第九圖中實線小視窗所示），並 疋義目標視窗大小。通常目標視窗尺寸設為7x7或 11χ11 (pixel)。 2. 如步驟1120所示’以理論切削路徑為圓心、，以設定 ，誤差半徑作-鄰近圓，其所涵蓋區域設為輪廊搜尋 範圍（如第九圖中大虛線圓所示）。 3. 如步驟1132與步驟1134所示，以一個像素增量方 式’ W叟尋範圍上方開始，以由上到下、由左到右方 f，設定搜尋範圍内所涵蓋的實際工件輪_匹配視 =t心’並綱見窗所涵蓋的實際工件輪扉設為匹配視 固（定義其尺寸與目標視窗相同，如第九圖中實線小 視窗所示）。 4·如A步驟m2、步驟1144、步驟祕與步驟1148所示， 計算目標視窗與匹配視窗之間的相似性參數。研究中 利用改良Candocia [60]的相似性參數公式作為兩視 窗間内的相似性參數計算之用。重複步驟3、4，當搜 尋範圍的匹配視窗皆搜尋完畢後，當相似性參數為區 1360029 域極大（為搜尋範圍内所計算的最大相似性參數) 時，此匹配視窗中心(實際工件輪廓）即為步驟丨目標 視固中心（理論切削位置）的最佳對應，亦即為實際 切削位置。 5.如步驟1150與步驟116〇所示，重複步驟卜2、3及 4直到所有理論切削位置皆搜尋到其最佳對應的座 標。 相似量測的相似性參數計算公式 ^ Candocia的相似量測參數計算式並無包含像素梯度的計 算’此易衍生誤判，因此在本發明之一較佳範例中將改以視窗 内特徵點與視窗巾^之間的練斜率來進行相似性參數計 算’同時$了避免誤認實際切削點，亦在相似性參數公式中另 2入了計算視窗巾战與前—鳩關座標的斜率關係式，以 曰升相似性判斷的可靠度。修正後的相似性參數計算式如下：The service match does not tear, _ causes the object _ path deviation _ first rule U the minimum position of the actual position Pa of the tool and the theoretical path when cutting. The track error (4) refers to the distance between the actual tool position Pa and the theoretical position Pd. When the mining error and tracking error are required, the actual position of the tool needs to be known. Because of the method of I I I, this method can _ 彳 true machining error '..., and the disadvantage 疋 can not be caused by the actual workpiece wheel gallery two kinds of errors (4) The actual processing position (Pa) and the corresponding difference and tracking error are required when the known theory is known, and therefore the method is most recognized in the present invention to cope with the above problem. In the error identification method, the knife is used to find the actual machining position of the tool in the contour matching method. _ehing ^ In the method, the small matching window will be used to correct the # (7) city shirt style, and the actual power local contour and similarity will be determined accordingly. The actual tool prosthesis of the theoretical tool position _ in the micro-tool machine has -^ reliable accuracy, by the next two concepts of fine ======= difference, in micro-contour distribution mode, two set =:= 1360029 face. (2). Within the tool cut. The true tool cutting point will fall on the error radius of the theoretical tool cutting point. Using the two assumptions mentioned above, when the theoretical domain point coordinates are obtained, it can be like turning the wealth __ spine. Define the image ί = the search conditions and the search target: (1) The second cut and the theoretical wheel age, when the tool cuts /, the theoretical contour has the shortest distance, define this technology ^ (2) The theory is cut _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The workpiece rim and Γ are defined as the feature points of the actual jade contour coordinate. The theoretical cutting point is set as the coordinate of the center of the target window. (7) According to the circle radius of the machine tool, the radius of the adjacent circle is converted into image resolution according to the image resolution. Like = early position, as the image matching principle of the search window search range 1 Candocia similarity parameter formula is used as the cut contour matching similarity calculation. When each theory _ point new search corresponds to the true point, According to the contour error and the tracking error in space (4), the error and tracking error after the reinforcement are analyzed. Taking the tenth figure as an example, ρι is the theoretical cutting point of the tool cutting path 而 and P2 is the actual cutting point. According to the contour error The principle finds P3 in the theoretical workpiece wheel red as the shortest distance between p2 and P2. The distance between P2 and P3 is the wheel coordinate error of the workpiece when the cutting path C1 is cut. E is based on the principle of tracking error, P2 to ρι The distance between the two indicates the tracking error of the cutting path C1 of the tool, E2. The feature matching principle of the shirt image is mainly to search for the relationship between the image and the empty object. The meaning of the relationship is as follows: Object point in the image plane , must shoot in the corresponding shadow plane to find its corresponding point. Tenth - the picture is the feature matching principle of the image stream private map 'steps are as follows: as shown in step 1112, step 1114 and step 1116, the theoretical cutting position is set It is the target window center, and the theoretical artifacts covered by the window, the target window (as shown in the solid-line small window in Figure 9), and the target window size. The target window size is usually set to 7x7 or 11χ11 (pixel). 2. As shown in step 1120, 'with the theoretical cutting path as the center, set with the error radius as the adjacent circle, and the area covered is set to the search range of the corridor (as shown by the large dashed circle in the ninth figure) 3. As shown in step 1132 and step 1134, start in the pixel increment mode 'W search range above, to set the actual workpiece wheel covered by the search range from top to bottom and left to right f. The matching workpiece =t heart's and the actual workpiece rim covered by the window is set to match the apparent solid (the size is defined to be the same as the target window, as shown in the solid small window in the ninth figure). 4. As the A step m2 Step 1144, step secret and step 1148, calculate the similarity parameter between the target window and the matching window. The similarity parameter formula of the modified Candocia [60] is used as the similarity parameter calculation between the two windows. Repeat steps 3 and 4. When the search window of the search range is searched, when the similarity parameter is the maximum of the zone 1360029 (the maximum similarity parameter calculated within the search range), the matching window center (actual workpiece contour) ) is the best correspondence of the target 视 target solid center (the theoretical cutting position), which is the actual cutting position. 5. Repeat steps 2, 3, and 4 as shown in steps 1150 and 116, until all theoretical cutting positions find their best corresponding coordinates. The similarity parameter calculation formula of similar measurement ^ Candocia's similar measurement parameter calculation formula does not contain the calculation of the pixel gradient 'this easy derivative decimation, so in a preferred example of the present invention will be changed to the feature points and windows in the window The slope between the wipes ^ is used to calculate the similarity parameter 'At the same time, to avoid misunderstanding the actual cutting point, and also to calculate the slope relationship between the window towel warfare and the front-marker coordinates in the similarity parameter formula. The reliability of the similarity judgment. The corrected similarity parameter is calculated as follows:
F(^+i) 其中 %、乂：視窗A、視窗B特徵點的個數 P 1 /办，為匹配權重（weight) hHn、NA，NB) 視窗A内’第q個特徵點距視窗B内最近特徵點的 19 1360029 & :視窗A内，坌Λ 率 特徵點與視窗甲心之間的座標斜 率知.視固B内’最近特徵點與視窗㈠心之間的座標斜 中心與前1論切削點之間的座標斜率 知·視窗B中心與前一真實切削點之間的座標斜率 斜率騎外職㈣正負把 測的：:二具;實施：係-雜 m 一刀具依據—數值控制碼於一工件上切 «取3ΓΪΓ320與步驟1330所示，以一圖糊取裝置 物·靖彡紐影像中 生-理論“。接下來==::，依據數_^^^ 从±人0 Α 步驟350所示，將工件輪廓與理 =輪靡疊合於同-座標系。之後，如步驟测與㈣所示， 定義理論輪射座標系上的複數伽論_點，並且定義工件 輪廟在座標系上的複數個真實_點。熟悉侧技術者可輕易 推知依據不㈣可定義料_格，複數倾論切削點盘 複數個真實切削點位於網格中的格子中。例如，網格可以是一 像素網格，每—理論_點與真實切削點分別對應於像素網格 中的一像素。 20 1360029 再接下來，如步驟1380所示，依據每一個理論切削點分 別找出與每-_論_點她_真實靖點…理論切削 點相匹配的真實切削點係於相應於理論切削點之一誤差範圍 内找出。在本發明之-範例中，更包含依據電腦紐控織台 之精確度定義-誤差半徑，其中誤差範圍為—圓，_以理論 切削點為圓心，並且以誤差半徑為半徑。 因此，依據每一個理論切削點分別找出與每一個理論切 削點相匹方法可包含：依據理論娜點定義 相應理論切削點的一理論視窗；依據誤差範圍内的每一個真實 切削點分別定義相應每-個真實切繼的—眺視窗；以及將 理論視窗與每-舰配視窗隨以找出與理論_點相匹配 的真實切繼。在本發明之-範财，紐論視窗與每一個匹 配視窗匹_ —a的立體雖匹配之她量測，以最相 似的匹配視窗所相應的真實切肖彳點作為理論切肖彳點相匹配的 真實切削點。 如先前所述，將理論視窗與每一個匹配視窗匹配係計算 理論視窗與每一個匹配視窗間的一相似性參數，其中理 似性參數為最小。其中ψμ —ΰ)=彳 β)為相似性參數； 點相匹配的真實切削點所相應的匹配視窗與理論視窗間的相 了-Η))F(^+i) where %, 乂: window A, window B feature points P 1 / do, for matching weight (weight) hHn, NA, NB) window A within 'the qth feature point from window B 19 1360029 & within the nearest feature point: inside the window A, the slope of the coordinate between the feature point and the center of the window is known. The center of the coordinate between the nearest feature point and the window (one) in the solid B 1 Discussion on the coordinate slope between the cutting points. The slope of the coordinate slope between the center of the window B and the previous real cutting point. Riding the external position (4) Positive and negative measurements:: Two; Implementation: Department-Miscellaneous m A tool basis - Value The control code is cut on a workpiece by taking 3ΓΪΓ320 and step 1330, and taking a picture to get the device and the image of Jingjing New Image-theory. Next ==::, according to the number _^^^ from ± people 0 Α As shown in step 350, the workpiece contour and the rim = rim are superimposed on the same-coordinate system. After that, as shown in step (4), the complex gamma _ point on the theoretical wheel coordinate system is defined, and the workpiece is defined. The plural real points of the temple in the coordinate system. Those familiar with the side can easily infer that the basis is not (4) The number of real cutting points of the cutting point disk is located in the grid in the grid. For example, the grid can be a pixel grid, and each of the theoretical_point and the real cutting point corresponds to a pixel in the pixel grid. 20 1360029 Next, as shown in step 1380, the true cutting point matching the theoretical cutting point of each of the theoretical cutting points is found according to each theoretical cutting point, and the error is corresponding to one of the theoretical cutting points. In the scope of the present invention, the definition is based on the accuracy of the computer control weaving table - the error radius, wherein the error range is - circle, _ is the center of the theoretical cutting point, and the radius of the error radius Therefore, finding a method corresponding to each theoretical cutting point according to each theoretical cutting point may include: defining a theoretical window of the corresponding theoretical cutting point according to the theoretical point; defining each true cutting point according to the error range Corresponding to each of the true-cut--windows; and the theoretical window and each-ship window to find the true succession that matches the theoretical point. In the present invention - Fancai The New Theory window is matched with the stereo of each matching window _a, but the real cut point corresponding to the most similar matching window is used as the actual cutting point of the theoretical matching point. As previously described, the theoretical window is matched with each matching window to calculate a similarity parameter between the theoretical window and each matching window, wherein the similarity parameter is the smallest. ψμ_ΰ)=彳β) is the similarity parameter. ; The matching point between the matching window and the theoretical window of the real cutting point matched by the point -Η))
\SA -5flL (-1) 沁、乂分別為理論視窗、匹配視窗中理論切削點 切削點個數； ‘ 户=1 //z，為一匹配權重； 21 1360029 办為凡與鸲兩者中最小的值； 近真2=^;’第q_論切削點距匹配視窗内一最 間的視糾’f q伽論蝴點與理論視窗中心之 的座= 匹配視窗内’最近真實切削點與匹配視窗中心之間 b +心與第q個理論_關前—理論切削 點之間的座標斜率；以及 知為匹配視窗中心與前一真實切削點之間的座標斜率。 如步驟1390所示，依據每一個理論切削點與相匹 配^、實關點分職別出相應於每—個理論域點的一加 ^差/、中加工决差包含一輪廓誤差與一循跡誤差，輪廟誤 差為理論__匹_真實切繼與理論輪廓_最短距 離，並且循祕差為理論切難相匹_真實_點與理論切 削點間的距離。 在本發明之-範例中’更包含依據數值控制石馬產生一刀 具切削路彳f ’理論切舰減於刀具_雜上的—刀具位 置，相應理論_點的加卫誤差為相應刀具位置的加工誤’差。 據此，本發明之另一具體實施例係一種一種機上二維卢 輪廓檢測的系、統，包含件、—内儲__數值控制碼的一電ς 數值控制機台與-誤差辯識裝置。其中數值控繼台依據 -數健綱.刀具n件上娜，糾，—誤差辯識裳 22 ljouuzy 為下列之& ^程搞之電腦，_之程式碼使得電腦成 F置含：以—圖形擷取裝置摘取工件之一影像的 Ϊ_測法依據影像產生影財的—工件輪 論輪廓叠合於同-座^的=輪裝f ’·將工件輪靡與理 直實:=ΓΤ定義工件輪_系上的複數個 理論切==^^=:_與每一個 ==轉_咖職減=:== iT兒明中=差的裝置。本具體實施例之其他細節已揭示於上 述說明中，在此不再贅述。 顯然地，錄上面實_巾的㈣，本發财能有許多 的6正與差異。因此需要在其附加的權利要求項之範圍内加以 ，解’除了上述詳細的描述外，本發明還可以廣泛地在其他的 實施例中施行。上龍為本發明之難實施_已，並非用以 限定本發明之”專利範圍；凡其它未脫離本判所揭示之精 神下所完成的等效改變或料，均應包含在下述巾料利範圍 内0 23 1360029 圖式簡單說明】 意圖 第一圖係為一機上, ‘維度輪廓檢測法流程示专圖. 第二圖係為機上二維度輪測法的相機與二方位示 第三A圖、第三B圖、第三c圖與第三〇 測法之範例比較示意圖； 圖為各邊緣檢 第四圖為校驗片之示意圖 第五A圖與第五B圖為二 值影像之邊界 示意 第六圖為理論J1件麵重賴程示意圖； 第七圖為實際工件輪廓與理論工件輪廓貼合流程圖； 第八圖為空間中循跡誤差與輪廓 二’ 墙丄^ 我之不意圖； 第九圖為微觀概念的局部輪廓示意圖； 第十圖為誤差模型示意圖；以及 第十一圖為影像匹配流程示意圖。 24 1360029 【主要元件符號說明】 CCD影像擷取裝置 Ο工件 E1輪廓誤差 E2循跡誤差 Pa實際位置 Pd理論位置 C1刀具切削路徑 P1理論切削點 P2實際切削點 P3理論輪廓上與實際切削點最相近點 25\SA -5flL (-1) 沁, 乂 are the theoretical window, the number of cutting points of the theoretical cutting point in the matching window; 'user=1 //z, which is a matching weight; 21 1360029 The smallest value; near true 2 = ^; 'q_ on the cutting point distance matching window in the innermost view of the 'fq gamma theory point and the center of the theoretical window = matching window inside 'the nearest real cutting point and Matches the slope of the coordinates between the b + heart and the qth theory _ off - the theoretical cut point between the center of the window; and the coordinate slope between the center of the matching window and the previous real cutting point. As shown in step 1390, according to each theoretical cutting point and the matching ^, the actual closing point, the corresponding difference corresponding to each theoretical domain point, the middle processing tolerance includes a contour error and a cycle Trace error, the wheel temple error is the theoretical __pi_true cut and the theoretical contour _ the shortest distance, and the secret difference is theoretically difficult to match the distance between the real_point and the theoretical cutting point. In the example of the present invention, 'there is a numerical control of the stone horse to produce a tool cutting path f 'theoretical cutting ship minus the tool - miscellaneous - the tool position, the corresponding theoretical _ point of the guard error is the corresponding tool position Processing error is poor. Accordingly, another embodiment of the present invention is a system, system, and component of an on-board two-dimensional lure contour detection, an electric ς numerical control machine and an error identification Device. Among them, the numerical control succession base is based on the number of health. The tool n pieces are on the na, the correction, the error is recognized by the singer 22 ljouuzy for the following & ^ procedure computer, _ the code makes the computer into F contain: to - The image capture device extracts the image of one of the workpieces. _ The measurement method produces the image based on the image. The contour of the workpiece wheel is superimposed on the same-seat ^ wheel mounted f '· The workpiece rim is straight and true: ΓΤDefining the number of theoretical rounds on the workpiece wheel _ system ==^^=: _ with each == turn _ _ _ _ = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Other details of this embodiment have been disclosed in the above description and will not be described again. Obviously, there are a lot of 6 positive differences between the present wealth and the (4). Therefore, it is intended that the present invention be construed as being limited to the scope of the appended claims.上 。 。 。 。 。 。 。 。 。 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ In the range 0 23 1360029 Simple description of the schema] The first image is intended to be on a machine, and the dimension of the dimension profile detection process is shown. The second image is the camera of the two-dimensional wheel test on the aircraft and the third orientation. A schematic diagram comparing the examples of the A picture, the third B picture, the third c picture and the third measurement method; the picture shows the fourth picture of each edge detection is a schematic diagram of the verification piece. The fifth A picture and the fifth B picture are binary images. The sixth figure shows the schematic diagram of the theoretical J1 surface weighting; the seventh picture shows the flow chart of the actual workpiece contour and the theoretical workpiece contour; The eighth picture shows the tracking error and contour in the space. The ninth figure is a schematic diagram of the partial contour of the microscopic concept; the tenth figure is the schematic diagram of the error model; and the eleventh figure is the schematic diagram of the image matching process. 24 1360029 [Description of main component symbols] CCD image capturing device Ο workpiece E1 contour Error E2 The actual cutting point and the nearest point 25 on the actual position error Pa Pd theoretical position of the tool cutting paths C1 theoretical cutting the actual cutting point P2 P3 theoretical profile points P1