# Remaining Useful Life Prediction of Cutting Tools Using an Inverse Gaussian Process Model
## 1 Introduction  
Tool wear is widely considered to be stochastic and challenging to predict. This is primarily due to unit-to-unit performance variations and process variations. Efficient approaches that can predict remaining useful life (RUL) are necessary for improving cutting quality and saving costs. According to the input data used in the performance degradation model, RUL prediction methods can be classified into three categories: time series models (with working time as input), artificial intelligence models (with real-time working data as input), and stochastic process models (with degradation data as input). The time series models analyze historical degeneration data to conduct RUL prediction using statistical approaches. Numerous time series models have been proposed and developed in tool RUL prediction in recent years, including the hidden Markov model [1, 2], the autoregressive integrated moving average model [3], Kalman filtering [4, 5], and particle filter [6, 7, 8, 9]. Methods based on artificial intelligence take the extracted signal features or the original signal as input and RUL as output. Many advanced artificial intelligence models have been widely used in tool residual life prediction, such as neural network [10, 11, 12], support vector machines [13], and deep learning methods [14, 15]. In stochastic processmodels, degradation over time is often modeled by a stochastic process \(\{Y(t);\,t\geq 0\}\) to account for damage accumulation, with inherent randomness. The RUL is determined as the first passage time of the process with respect to some failure threshold. In this context, three types of degradation modelling technique are widely discussed in the literature, namely the Wiener process model [16; 17; 18], the gamma process model [19], and the inverse Gaussian process model [20; 21]. Meanwhile, [PERSON] and [PERSON] combined neural networks and image processing for tool life prediction [22]. The time series models are suitable for mass production, with abundant historical degeneration data, while the artificial intelligence methods are appropriate for dealing with massive and complicated process data. Both the time series methods and the artificial intelligence methods are based on the invariable degradation trajectory. However, the tool wear process is complicated by randomness and periodicity, which are related to friction speed, pressure, surface roughness, material properties, friction and wear types, lubrication status, surface coating, and individual differences between tools. All of these factors lead to uncertainty in the tool degradation process. Therefore, it is more reasonable to describe tool performance degradation using stochastic processes. Although tools of the same type have commonalities in design and material, there might be significant individual differences due to dynamic use conditions. To characterize individual differences, the random-effects model is introduced in the stochastic process model [23]. [PERSON] and [PERSON] [24] introduced a random variable into the degradation model to describe individual differences. [PERSON] and [PERSON] [25] imposed a random effect on the drift parameter of the Wiener process, where a normal distribution is assumed for the random drift across the population. Compared to the Wiener process model and the gamma process model, the IG process is flexible in incorporating random effects that account for heterogeneities commonly observed in degradation problems [26; 27]. In the current cutting process study, precision is generally used to describe the precision level of machine tool [28]. In the actual cutting process, the precision of the product is not only related to the precision of the machine, but also related to the level of the operator, the state of the cutting tool, the state of the fixture, the material characteristics of the cutting workpiece, and the processing technology. The whole machining system will affect the precision. In this paper, the research object is the cutting tool.