# Real-Time Estimation for Cutting Tool Wear Based on Modal Analysis of Monitored Signals
## 2 Algorithm Theory and Method
### Monitored Signals and Tool Wear  
Signal monitoring is of vital importance in order to estimate tool wear. The researches need to get sufficient monitoring information of tool condition. Cutting force can be used on condition estimation [15; 16] as it is directly related to wearing process, meanwhile, some researchers payed more attention on vibration signals to estimate tool wear [17; 18]. The precision of tool wear that obtained through directly measuring tool is high but affects the production, and it is also not easy to implement the on-the-fly measurement of tool wear in the actual production. Therefore, research on indirect measuring methods is necessary, and tool wear is analyzed by measuring its relationship for milling force, vibration signals, AE signals, etc. To determine the best information for tool condition analysis, investigation is conducted to obtain the related parameters for condition analysis. Correlation analysis is investigated to observe whether there is some kind of interdependent relationship between these signals and tool wear.  
If \(X\) is a random variable, \(\mathrm{E}(X)\) is the expectation of \(X\) and \(\mathrm{E}\{\left[X-\mathrm{E}(X)\right]^{2}\}\) is exist, \(\mathrm{E}\{\left[X-\mathrm{E}(X)\right]^{2}\}\) will be called the variance of \(X\), marked as \(\mathrm{D}(X)\). \(\mathrm{E}\{\left[X-\mathrm{E}(X)\right][Y-\mathrm{E}(Y)]\}\) is called the covariance of the random variable \(X\) and \(Y\), marked as \(\mathrm{Cov}(X,Y)\). The correlation coefficient \(\rho_{xy}\) of \(X\) and \(Y\) is defined as:  
\[\rho_{xy}=\frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{D}(X)}\sqrt{\mathrm{D}(Y)}} \tag{1}\]  
In which, \(\mathrm{D}(X)\) is the variance of \(X\) and \(\mathrm{D}(Y)\) is the variance of \(Y\).  
The linear function \(a+bX\) of \(X\) is used to approximate \(Y\), and the mean square error \(\mathrm{e}\) is used to measure the degree of goodness of \(Y\).  
\[\begin{split} e=\mathrm{E}\Big{[}(Y-(a+bX))^{2}\Big{]}\\ =\mathrm{E}\Big{(}Y^{2}\Big{)}+b^{2}E\Big{(}X^{2}\Big{)}+a^{2}-2 b\mathrm{E}(XY)+2 ab\mathrm{E}(X)-2a\mathrm{E}(Y)\end{split} \tag{2}\]  
The smaller the value of \(e\) is, the greater the approximate degree of \(a+bX\) and \(Y\) is. Find the partial derivatives of \(e\) about \(a\) and \(b\) and make them equal to zero.  
\[\left\{\begin{array}{l}\frac{\partial e}{\partial a}=2a+2b\mathrm{E}(X)-2 \mathrm{E}(Y)=0\\ \frac{\partial e}{\partial b}=2b\mathrm{E}\big{(}X^{2}\big{)}-2\mathrm{E}(XY)+ 2a\mathrm{E}(X)=0\end{array}\right. \tag{3}\]  
The solutions as follows:  
\[b_{0}=\frac{\mathrm{Cov}(X,Y)}{\mathrm{D}(X)} \tag{4}\]  
\[a_{0}=\mathrm{E}(Y)-b_{0}\mathrm{E}(X)=\mathrm{E}(Y)-\mathrm{E}(X)\frac{ \mathrm{Cov}(X,Y)}{\mathrm{D}(X)} \tag{5}\]\(a_{0}\) and \(b_{0}\) will be substituted into (6):  
\[\text{minE}\Big{[}(Y-(a+bx))^{2}\Big{]}=\text{E}\Big{[}(Y-(a_{0}+b_{0}X))^{2} \Big{]}=\Big{(}1-{\rho_{xy}}^{2}\Big{)}\text{D}(Y) \tag{6}\]  
It can be seen from the Formula (6) that \(\rho_{xy}\leq 1\), and \(\rho_{xy}\) can show the linearity between \(X\) and \(Y\). When \(\rho_{xy}\) is large, the correlation degree between \(X\) and \(Y\) is better; when \(\rho_{xy}\) is small, it shows that the correlation degree between \(X\) and \(Y\) is poor; when \(\rho_{xy}=1\), there is a linear relationship between \(X\) and \(Y\); when \(\rho_{xy}=0\), it is said that \(X\) and \(Y\) are not related.  
The monitored signals showing the high correlation with tool wear are chosen to extract features.