- 學位論文
快速NURBS路徑插補及幾何誤差補償方法
Fast NURBS path interpolation and geometric errors compensation
摘要
電腦數值控制工具機現在已經成為各種製造工業裡最主要的生產力提供者。面對目前的市場環境，產品製造商在維持高生產力的同時又得兼顧產品的品質下背負了龐大的壓力。為了維持一定的產品精度，機器所造成體積誤差必須校正與補償以避免製造出有缺陷的產品。為了迎合消費者喜好，產品的幾何形狀設計越來越複雜，設計人員也越來越習慣使用自由曲線與曲面來設計新的產品。NUBRS曲線與曲面定義近來被廣泛應用在CAD/CAM系統，NURBS格式定義對於工件幾何形狀的描述遠比傳統CAM系統用後處理器產生大量小直線線段來逼近的精度高。另外由於NURBS格式的檔案資料量遠比線段逼近小，因此傳輸上負擔比較輕，加上單一路徑長度足夠讓機器加速到較高的切削速度，使得複雜曲面的高速切削加工可行性大為提高，許多高階的CNC控制器製造商已經將NURBS插補功能視為標準模組。因此，開發一個具有NURBS插補功能並整合體積誤差補償的CNC控制器將會對產品製造商帶來莫大的幫助。 本論文提出一個快速的NURBS路徑插補機制。該插補機制將CNC控制器中NURBS路徑資料從預處理到即時插補模組進行了有效率的整合。NURBS路徑的長度計算採用了適應面積法(adaptive quadrature)對長度函數的一次微分進行積分，該方法根據設定的計算精度會自動將曲線分割成許多區段，數量多寡取決於曲線的複雜度。本論文利用細分區段的結果來建構曲線長度逆轉換函式(Inverse length function)，進行即時插補計算時可直接利用曲線長度逆轉換函式計算得到相關曲線位置參數，大幅地減輕了即時插補計算的負擔。本方法可快速且準確地提供NURBS即時插補運算，模擬與實際切削結果也驗證了該插補機制的效果。 此外，本論文也提出一個對於具NURBS插補功能CNC工具機進行幾何誤差補償的快速方法。該方法利用原始路徑的NURBS基底函數來估算實際機器運動時所造成的誤差函數，由於誤差函數使用與原始路徑同樣的NURBS基底函數，因此路徑修正時只需要修正原始路徑的控制點即可達到補償的目的。如果原始路徑的基底函數無法準確描述誤差函數，則可利用NURBS節點插入法來增加原始路徑的易曲性，也同時改善本方法的補償精度。本方法主要將補償功能實現在開放式架構CNC控制器中，雖然不是在即時位置控制模組直接進行定位補償，但在NURBS插補器之前進行補償處理，在適當控制資料區塊讀取條件下，機台的一些即時狀態可在補償過程中列入考慮，因此本方法可視為一種on-line補償機制。本方法不但具有on-line補償的優點，同時所有計算都在非即時模組中完成，完全不會增加即時控制模組的計算負擔。模擬與實驗量測結果均驗證了本方法可以有效的將機器幾何誤差有效的補償到要求精度範圍內。本方法同時也可以實現在CAM系統的後處理器中，但不具備相關on-line補償的特性。
並列摘要
Computer numerical control (CNC) machine tools have now become the primary production tool for all kinds of manufacturing industries. Manufacturers are under tremendous pressure to enhance product quality in terms of dimension accuracy while maintaining high productivity. To maintain product quality, the volumetric errors of machine tools should be calibrated or eliminated in advance to prevent the manufacture of defective parts. As modern products adopt more complex geometric shapes, it is becoming common for designers to employ free-form curves and surfaces. Non-uniform rational B-spline (NURBS) curves and surfaces are widely used in CAD/CAM systems. The NURBS formats have much more precise mold shape descriptions than conventional CAM approximations with tiny line segments. The data transfer burden is reduced due to the smaller NC data file with the NURBS format. The real cutting speed is increased and is stable for sufficient path length in the feedrate profile scheduler. All these benefits are important for high-speed cutting programming. The NURBS interpolator has recently become the standard module for the latest high-performance CNC controllers. Therefore, the designing and building of a CNC controller with NURBS interpolation and volumetric error compensation functions will be of great help to manufacturers. In this thesis, a fast real-time NURBS path interpolation method is proposed. The proposed method efficiently integrates the data processing of a NURBS path in a CNC controller, from pre-processing to real-time interpolation. In the calculation of the total length of the NURBS path, the numerical adaptive quadrature method adapts to the integrand, i.e. the first derivative of the length function, automatically, dividing the parameter interval into subintervals with fine or coarse spacing according to the varying condition of the integrand. This new method takes full advantage of the subdivision scheme. The key point is to generate inverse length functions (ILF) for each resulting subinterval. In the real-time NURBS path interpolation, the new setting path parameter can be calculated directly using the inverse length function without any time-consuming computation for NURBS parameter approximation. The proposed method is extremely fast, accurate and suitable for real-time implementation, and simulations and a practical cutting test have proved its effectiveness. Another important emphasis in this thesis is to present a new scheme for compensating the geometric errors of a CNC machine tool which is capable of NURBS interpolation. The proposed NURBS-based compensation method utilizes the same basis functions as the setting tool path to directly estimate the actual NURBS path. Due to the use of the same basis functions, the modification of control points is a fast and effective way to compensate the setting tool path. The knot insertion technique of NURBS is used to increase the flexibility and enhance the accuracy of compensation at the same time. This thesis proposes a fast method of on-line compensation of the total positioning errors of a setting NURBS tool path in advance without any additional real-time CPU loading. Both the simulation and experimental results show that the positioning deviations can be effectively enhanced using the fast geometric error compensation method. This proposed method can be also implemented in the post-processor of a CAM system for off-line compensation.