# Three-Stage Wiener-Process-Based Model for Remaining Useful Life Prediction of a Cutting Tool in High-Speed Milling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three-Stage Wiener-Process-Based Model

#### 2.1. Motivation

#### 2.2. Model Formulation

## 3. RUL Prediction Framework

#### 3.1. Feature Extraction and Selection

#### 3.1.1. Feature Extraction

#### 3.1.2. Automatic Feature Selection with SDAE

#### 3.2. Tool Wear Stage Classification and Health Indicator Construction

#### 3.3. Parameter Estimation and RUL Prediction

#### 3.3.1. Offline Estimation of Initial Model Parameters

#### 3.3.2. Online Updating of Model Parameters

**E-Step:**Calculating the expectation of log-likelihood function based on the $jth$ iteration.

**M-Step:**Calculating the parameter ${\widehat{\Phi}}_{}^{(j+1)}$ in the $(j+1)th$ iteration.

#### 3.3.3. RUL Prediction

- PDF of lifetime

- PDF of RUL up to ${t}_{k}$

## 4. Experimental Study

#### 4.1. Experiment Set-Up and Data Acquisition

#### 4.2. Feature Selection with SDAE

#### 4.3. Experimental Results

#### 4.3.1. Model Parameter Estimation

- (1)
**Stage-1**$$u(t;\theta )=a{t}^{b}$$- (2)
**Stage-2**$$u(t;\theta )=a{t}^{b}+c\mathrm{exp}(dt)-c$$- (3)
**Stage-3**$$u(t;\theta )=c\mathrm{exp}(dt)-c$$

Model Parameters | Cutter #1 | Cutter #4 | Cutter #6 | ||||||
---|---|---|---|---|---|---|---|---|---|

Stage-1 | Stage-2 | Stage-3 | Stage-1 | Stage-2 | Stage-3 | Stage-1 | Stage-2 | Stage-3 | |

$a$ | 31.7602 | 89.7611 | ― | 23.4435 | 52.1428 | ― | 38.1735 | 89.0511 | ― |

$b$ | 0.2582 | 0.0001 | ― | 0.3071 | 0.1082 | ― | 0.2291 | 0.0462 | ― |

$c$ | ― | 3.9420 | 122.8121 | ― | 3.0810 | 89.6324 | ― | 50.1321 | 128.2356 |

$d$ | ― | 0.0144 | 0.0031 | ― | 0.0082 | 0.0073 | ― | 0.0019 | 0.0044 |

${\beta}_{0}$ | 0.0107 | 0.0969 | 0.0925 | 0.0108 | 0.0145 | 0.0283 | 0.0028 | 0.0173 | 0.0543 |

${\beta}_{1}$ | 0.9837 | 1.1215 | 0.9995 | 1.0031 | 0.9998 | 0.9999 | 1.0001 | 0.9986 | 0.9998 |

${\sigma}_{B}$ | 0.5046 | 0.1312 | 0.4480 | 0.4612 | 0.1678 | 0.9885 | 0.9715 | 0.1150 | 0.7297 |

$\gamma $ | 0.0140 | 0.0133 | 0.0436 | 0.0135 | 0.0138 | 0.0324 | 0.0145 | 0.0135 | 0.0397 |

#### 4.3.2. RUL Prediction Result

#### 4.3.3. Comparison and Evaluation

- (1)
- MSE$$MSE=\frac{1}{K}{\displaystyle \sum _{k=1}^{K}{\left({l}_{k}-{\tilde{l}}_{k}\right)}^{2}}$$
- (2)
- MAE$$MAE=\frac{1}{K}{\displaystyle \sum _{k=1}^{K}\left|{l}_{k}-{\tilde{l}}_{k}\right|}$$
- (3)
- ${R}^{2}$$${R}^{2}=1-\frac{{\displaystyle {\sum}_{k=1}^{K}{\left({l}_{k}-{\tilde{l}}_{k}\right)}^{2}}}{{\displaystyle {\sum}_{k=1}^{K}{\left({l}_{k}-\overline{l}\right)}^{2}}}$$

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Three stages of tool wear in a cutting process. (

**a**) The typical degradation process of a cutting tool; (

**b**) Slight wear stage; (

**c**) Medium stage; (

**d**) Severe wear stage.

**Figure 4.**Raw data of Cutter #1. (

**a**) Vibration signal in X direction; (

**b**) Vibration signal in Y direction; (

**c**) Vibration signal in Z direction; (

**d**) Cutting force signal in X direction; (

**e**) Cutting force signal in Y direction; (

**f**) Cutting force signal in Z direction; (

**g**) Acoustic emission signal; (

**h**) True wear value.

**Figure 6.**RUL prediction results with proposed model and other existing models. (

**a**) Cutter #1; (

**b**) Cutter #4; (

**c**) Cutter #6.

**Figure 9.**Comparison of RULs and estimation errors between our model and power model. (

**a**) Cutting pass 230; (

**b**) Cutting pass 240; (

**c**) Cutting pass 250; (

**d**) Cutting pass 260; (

**e**) Cutting pass 270; (

**f**) Cutting pass 280; (

**g**) Cutting pass 290; (

**h**) Cutting pass 300; (

**i**) Cutting pass 310.

Step 1: Set the parameters $\theta ,{\sigma}_{B},\xi ,\gamma $ |

Step 2: Estimate the state ${\widehat{S}}_{k\left|k-1\right.}$ and variance ${P}_{k\left|k-1\right.}$${\widehat{S}}_{k\left|k-1\right.}={\widehat{S}}_{k-1\left|k-1\right.}+{\displaystyle {\int}_{0}^{{t}_{k}}g(\tau ;\theta )d\tau -{\displaystyle {\int}_{0}^{{t}_{k-1}}g(\tau ;\theta )d\tau}}$ ${P}_{k\left|k-1\right.}={P}_{k-1\left|k-1\right.}+{\sigma}_{B}^{2}\Delta t$ |

Step 3: Calculate the Kalman coefficient $K(k)$$K(k)={P}_{k\left|k-1\right.}/({P}_{k\left|k-1\right.}+{\gamma}^{2})$ |

Step 4: Update the state and variance${\widehat{S}}_{k\left|k\right.}={\widehat{S}}_{k\left|k-1\right.}+K(k)({O}_{k}-\varphi ({S}_{k};\xi ))$ ${P}_{k\left|k\right.}=(1-K(k)){P}_{k\left|k-1\right.}$ |

Step 1: Forward iteration through Kalman filter and obtain the optimal estimation ${\widehat{S}}_{k\left|k\right.}$ and ${P}_{k\left|k\right.}$. |

Step 2: Optimal smoothing estimation of backward iteration${R}_{j}={P}_{j\left|j\right.}{P}_{j+1\left|j\right.}^{-1}$ ${\widehat{S}}_{j\left|k\right.}={\widehat{S}}_{j}+{R}_{j}({\widehat{S}}_{j+1\left|k\right.}-{\widehat{S}}_{j+1\left|j\right.})={\widehat{S}}_{j}+{R}_{j}({\widehat{S}}_{j+1\left|k\right.}-{\widehat{S}}_{j})$ ${P}_{j\left|k\right.}={P}_{j\left|j\right.}+{R}_{j}^{2}({P}_{j+1\left|k\right.}-{P}_{j+1\left|j\right.})$ |

Step 3: Initialization |

${M}_{k\left|k\right.}=\left(1-{K}_{k}\Delta t\right){P}_{k-1\left|k-1\right.}$ |

Step 4: Smoothing covariance calculation of backward iteration${M}_{j\left|k\right.}={P}_{j\left|j\right.}{S}_{j-1}+{S}_{j}({M}_{j+1\left|k\right.}-{P}_{j\left|j\right.}){S}_{j-1}$ |

Feature | Expression | Feature | Expression |
---|---|---|---|

T1 | $\mathrm{M}={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^{T}f(t)}$ | T9 | $KR=\frac{{\displaystyle {\sum}_{t=1}^{T}f{(t)}^{4}}}{\sqrt{{\displaystyle {\sum}_{t=1}^{T}f{(t)}^{2}}}}$ |

T2 | $\mathrm{STD}=\sqrt{{\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^{T}{(f(t)-M)}^{2}}}$ | T10 | $ARV={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^{T}\left|f(t)\right|}$ |

T3 | $VAR={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^{T}{(f(t)-M)}^{{}^{2}}}$ | T11 | $\mathrm{SF}=\frac{RMS}{ARV}$ |

T4 | $P=\mathrm{max}(f(t))$ | T12 | $CP=\frac{PP}{\mathrm{RMS}}$ |

T5 | $\mathrm{PP}=\mathrm{max}\left(\mathrm{f}\right(\mathrm{t}\left)\right)-\mathrm{min}\left(\mathrm{f}\right(\mathrm{t}\left)\right)$ | T13 | $\mathrm{IF}=\frac{\mathrm{PP}}{ARV}$ |

T6 | $\mathrm{RMS}=\sqrt{{\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^{T}{(f(t))}^{2}}}$ | T14 | $CL=\frac{\left|PP\right|}{{\left({\displaystyle \frac{1}{T}}\sqrt{{\displaystyle {\sum}_{t=1}^{T}\left|f(t)\right|}}\right)}^{2}}$ |

T7 | $\mathrm{S}K=\frac{\frac{1}{T}{\displaystyle {\sum}_{t=1}^{T}{(f(t)-\mathrm{M})}^{3}}}{ST{D}^{3}}$ | T15 | $CS=\frac{{\displaystyle {\sum}_{t=1}^{T}f{(t)}^{3}}}{M}$ |

T8 | $\mathrm{KU}=\frac{\frac{1}{T}{\displaystyle {\sum}_{t=1}^{T}{(f(t)-M)}^{4}}}{ST{D}^{4}}$ | T16 | $CK=\frac{{\displaystyle {\sum}_{t=1}^{T}f{(t)}^{4}}}{M}$ |

Feature | Expression | Feature | Expression |
---|---|---|---|

F1 | $PC={\displaystyle \sum _{i=1}^{n}{p}_{i}{}^{2}}$ | F5 | ${M}_{p}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}S{(f)}_{i}}$ |

F2 | $FC=\frac{{\displaystyle {\sum}_{i=1}^{n}{f}_{i}{p}_{i}}}{{\displaystyle {\sum}_{i=1}^{n}{p}_{i}}}$ | F6 | $VA{R}_{p}=\frac{{\displaystyle {\sum}_{i=1}^{n}{(S{(f)}_{i}-{M}_{p})}^{2}}}{n-1}$ |

F3 | $VF=\frac{{\displaystyle {\sum}_{i=1}^{n}{({f}_{i}-FC)}^{2}{p}_{i}}}{{\displaystyle {\sum}_{i=1}^{n}{p}_{i}}}$ | F7 | $K{U}_{p}={\displaystyle \frac{1}{n}}\frac{{\displaystyle {\sum}_{i=1}^{n}{(S{(f)}_{i}-{M}_{p})}^{4}}}{VA{R}_{p}{}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ |

F4 | $MSF=\frac{{\displaystyle {\sum}_{i=1}^{n}{f}_{i}{}^{2}{p}_{i}}}{{\displaystyle {\sum}_{i=1}^{n}{p}_{i}}}$ | F8 | $S{K}_{p}={\displaystyle \frac{1}{n}}\frac{{\displaystyle {\sum}_{i=1}^{n}{(S{(f)}_{i}-{M}_{p})}^{3}}}{VA{R}_{p}{}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ |

Model | Cutter #1 | Cutter #4 | Cutter #6 | |||
---|---|---|---|---|---|---|

MSE | MAPE (%) | MSE | MAPE (%) | MSE | MAPE (%) | |

Our model | 42.80 | 4.78 | 22.76 | 5.19 | 48.64 | 7.01 |

M1 | 9355.50 | 35.85 | 1367.30 | 41.99 | 8867.40 | 28.69 |

M2 | 492.39 | 11.85 | 569.57 | 12.66 | 462.39 | 9.20 |

M3 | 457.76 | 13.96 | 382.71 | 7.99 | 528.68 | 13.27 |

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**MDPI and ACS Style**

Liu, W.; Yang, W.-A.; You, Y.
Three-Stage Wiener-Process-Based Model for Remaining Useful Life Prediction of a Cutting Tool in High-Speed Milling. *Sensors* **2022**, *22*, 4763.
https://doi.org/10.3390/s22134763

**AMA Style**

Liu W, Yang W-A, You Y.
Three-Stage Wiener-Process-Based Model for Remaining Useful Life Prediction of a Cutting Tool in High-Speed Milling. *Sensors*. 2022; 22(13):4763.
https://doi.org/10.3390/s22134763

**Chicago/Turabian Style**

Liu, Weichao, Wen-An Yang, and Youpeng You.
2022. "Three-Stage Wiener-Process-Based Model for Remaining Useful Life Prediction of a Cutting Tool in High-Speed Milling" *Sensors* 22, no. 13: 4763.
https://doi.org/10.3390/s22134763