精密工業中，進行加工過程的工件經常是相當昂貴的，因此對於品質的要求更是嚴謹，希望能以外力的補償上達到一定的水準；然而在摩擦力的補償往往會受到「遲滯效應」影響，過去文獻針對此效應提出相關數學模型，但實務上卻因受限於不同機型與工況導致模型無法良好建立進而進行完整的補償。
本研究提出一套針對遲滯效應區段的兩階段摩擦力模型架構與參數估計，應用於精密機器的伺服控制系統。第一階段，本研究根據物理模型的假設與多元線性迴歸分析(Multiple Linear Regression)進行機台摩擦力的數值估計，並透過馬達速度變化對於摩擦力影響之實驗，找出影響其大小的重要變數為加工位置的變化；接著，透過觀察摩擦力相對位置變化的關係找出數學模型，其結果與相關文獻的內容相互呼應；最後，利用自我迴歸模型(Autoregressive Model)與訊號處理中的Z轉換(Z-Transform)進行數學模型中的參數估計；另一方面，針對不同機台與加工條件，根據第一階段所建立的基礎模型於第二階段透過符號式迴歸(Symbolic Regression)進行殘差分析，並以基因編程(Genetic Programming)的手法完成符號式迴歸。
本研究以台灣電子設備製造公司為例進行實證研究，以臥式鑽孔中心機實驗並透過電子循圓測試的方式，來觀察因遲滯效應而產生的循圓尖角誤差藉由本研究第一階段提出的基礎摩擦力模型所補償之結果；另一方面，透過預測三種不同的CNC機台之摩擦力驗證此研究提出的兩階段摩擦力預測與補償架構能適用於各式機台與工況，成果顯示此架構所建構之摩擦力模型皆能達到消彌循圓尖角誤差以及估測CNC機台摩擦力的效果，進一步保持精密加工的精度與減少裝設感測器的成本，並有利於設備的故障預測和健康管理。

In the precision industry, the quality requirements of work pieces are strict. It is hoped that the compensation of the external force will reach a certain level. However, the compensation of the friction is often subject to the hysteresis effect which is nonlinear phenomenon.
This study proposes a two-phase framework of friction model and parameter estimation for the hysteresis effect segment, which is applied to the servo control system of precision machines. In the first phase, based on the assumptions of the physical model and the multiple regression analysis, the numerical estimation of the friction of the machine is carried out. Through the experiment, the important variable affecting the friction is discovered as the change of the machining position. Then, the mathematical model is initial found observing the relationship between the frictional force and the displacement; moreover, the results correspond to the related literature. Finally, autoregressive model and Z-transform are used to estimate the parameters in the model. On the other hand, for different machines and processing conditions, this study, in the second phase, uses symbolic regression to conduct residual analysis based on the exponential-based model established in the first phase, and genetic programming is applied to complete symbolic regression.
This study takes electronic equipment manufacturing company in Taiwan as an example to conduct an empirical study. The tapping center machine experiment and the electronic circle test method are used to observe the circular spike error caused by the friction force to validate the proposed exponential-based model in the first phase. Furthermore, for predicting the friction of three different CNC machines, the proposed two-phase friction prediction and compensation framework can be applied to various machines and working conditions. The result shows that the friction models constructed by this framework can achieve the effect of eliminating the circular spike error and estimating the friction of the CNC machines further maintain the precision of machining and save the cost of additional sensors. The result benefits the equipment prognostic and health management (PHM).

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