methods [6–8] can be applied to automatically search for key SFs to reduce the number of SFs during the model‐building and model‐refreshing processes so as to improve the model accuracy. However, due to the dynamic nature of these methods, the content of key SFs could vary after applying automatic search in each model refreshing, which might not be appropriate for implementation considerations.
For easy implementation, a fixed and concise set of SFs is required to represent the significance of the entire manufacturing process. Therefore, an expert‐knowledge‐based (EK‐based) selection procedure to find a fixed and concise set of SFs is illustrated below.
EK‐based Selection Procedure
In view of selecting the SFs of a vibration sensor (i.e. accelerometer), since the machining quality is affected directly by the tool states, SFs that can accurately monitor tool status should be selected. In high‐speed machining operations, a serious increase in cutting energy generated due to tool breakage or flank wear will amplify vibration magnitude that can be detected by the max, RMS, and avg of the vibration signal. These three SFs are crucial to the detection of vibration amplitude and energy variance between workpieces and tools.
A rolling bearing is one of the most important components widely embedded in machine tools. Therefore, abnormal statuses of rolling bearings may also cause breakdowns in rotating systems and result in serious machining failures. skew and kurt are two useful SFs to detect rolling bearing faults at an early stage. Besides, kurt, which works very well in the whole range from slow to very fast rolling speed, is sensitive enough to provide rich information of incipient faults for characterizing the impact existing in the rolling bearings. Further, std, a kind of precision‐related SF responsible for investigating any small changes during machining, can indicate good correlation with real precision values of workpieces.
Then, the essential and concise SFs of electric‐current signals are investigated. RMS of spindle current can correctly represent dynamic cutting‐force variation for monitoring tool fracture and precision prediction. When dealing with the alternating current (AC), CF is applied for detecting whether an electrical system has the ability to generate a particular current output. In addition, avg can also be used as an SF of tool flute breakage or tool‐wear estimations.
Finally, with the same reason as for vibration signals, max is used for detecting any abnormal current peaks during machining. Accelerometer and current sensors are selected here to demonstrate how to choose essential and concise SFs. By the same token, the essential and concise SFs of other sensors, such as dynamometers, acoustic‐emission sensors, and thermometers, can also be identified. In summary, the six vibration SFs selected include: max, RMS, avg, skew, kurt, and std; while the four current SFs chosen are RMS, avg, max, and CF.
Cross‐Correlation SFs
Generally, the Pearson product‐moment correlation coefficient [13] measures the similarity degree between two signals in the same time without any time lag. Given that x(t) and y(t) are two continuous‐time signals with time T, and their correlation coefficient value crxy, ranging from negative one to positive one, is defined in (2.7).
The cross‐correlation is similar to correlation coefficient, but it takes time lag into consideration. One signal is allowed to be time‐shifted and slide over the other to compare the similarity of two independent signals at each stride. It helps to find out where the two waveforms match the best at a certain time.
For random signals, the cross‐correlation CRxy between x(t) and y(t) is expressed in (2.8):
where γxy is the covariance expressed in (2.9):
Thus, cross‐correlation can be simplified by the ratio in (2.10):
For deterministic signals, the cross‐correlation of two continuous and periodic signals x(t) and y(t) can be defined by the integration from +∞ to −∞ as in (2.11), where notation * denotes the complex conjugate; x(t) is fixed and y(t) is shifted forward/backward by m, which is the displacement, or the so‐called lag.
For discrete signals x[n] and y[n] with length N, the cross‐correlation is defined as in (2.12), the products of two signals and the integration are replaced by any interval of period T at each point.
Cross‐correlation repeats to successively slide one signal along the x‐axis and compare with the other until the maximized correlation value is found. The reason is that two signals with the same sign (both positive or negative) tend to have a large correlation. Especially, when both peaks or troughs are aligned, it must be the best correlation. On the other hand, when signals have opposing signs at a certain time, its correlation or integral area must be small.
Cross‐correlation is very useful in the pattern recognition within a signal or between two signals. It is widely used to check the stability of sensor data and remove noise in a mass production environment. Note that, each CRxy can serve as a critical SF in a set, which can be expressed as SFCR(xy) = CRxy.
Autocorrelation SFs
Autocorrelation, or the so‐called serial correlation, performs the same cross‐correlation procedure of a signal with the time‐shifted form of itself. Thus, all autocorrelation has to do is to replace y(t) with x(t) from (2.8) to (2.12).
