Topological feature vectors for chatter detection in turning processes

Abstract

Machining processes are most accurately described using complex dynamical systems that include nonlinearities, time delays and stochastic effects. Due to the nature of these models as well as the practical challenges which include time-varying parameters, the transition from numerical/analytical modeling of machining to the analysis of real cutting signals remains challenging. Some studies have focused on studying the signals of cutting processes using machine learning algorithms with the goal of identifying and predicting undesirable vibrations during machining referred to as chatter. These tools typically decompose the signal using Wavelet Packet Transforms (WPT) or Ensemble Empirical Mode Decomposition (EEMD). However, these methods require a significant overhead in identifying the feature vectors before a classifier can be trained. In this study, we present an alternative approach based on featurizing the time series of the cutting process using its topological features. We first embed the time series as a point cloud using Takens embedding. We then utilize Support Vector Machine, Logistic Regression, Random Forest and Gradient Boosting classifier combined with feature vectors derived from persistence diagrams, a tool from persistent homology, to encode chatter’s distinguishing characteristics. We present the results for several choices of the topological feature vectors, and we compare our results to the WPT and EEMD methods using experimental turning data. Our results show that in two out of four cutting configurations the Topological Data Analysis (TDA)-based features yield accuracies as high as 97%. We also show that combining Bézier curve approximation method and parallel computing can reduce runtime for persistence diagram computation of a single time series to less than a second thus making our approach suitable for online chatter detection.

Appendices

Appendix A. Expressions for persistence paths’ signatures

Let the k th landscape functions be λ_k(x) = y where x and y represent coordinates along the birth time and the persistence axes. Since the persistence landscapes are piecewise linear functions, we can write them in closed form in terms of the nodes $\{x_{i}, y_{i}\}_{i=1}^{n}$ that define the boundaries of each of their linear pieces according to

$$ \lambda_{k}(t) = \begin{cases} \frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}} t + \frac{y_{i+1}(x_{i+1}-x_{i})+x_{i+1}(y_{i}-y_{i+1})}{x_{i+1}-x_{i}}, & \text{ for }i \in \{1,2,\ldots, n\} \text{ and } t \in [x_{1}, x_{n}], \\ 0 , &\text{ otherwise }. \end{cases} $$

(9)

The corresponding path is given by

$$ \begin{array}{@{}rcl@{}} P&=&[{P_{t}^{1}},{P_{t}^{2}}] = \left[t,\frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}} t \right.\\&&\left.+ \frac{y_{i+1}(x_{i+1}-x_{i})+x_{i+1}(y_{i}-y_{i+1})}{x_{i+1}-x_{i}}\right], \end{array} $$

(10)

and its differential is

$$ dP = [d{P_{t}^{1}},d{P_{t}^{2}}] = [dt, \frac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}} dt]. $$

(11)

Using the above definitions, we can derive general expressions for the first and second level signatures, respectively, according to

$$ S_{1} = {\int}_{x_{1}}^{x_{n}} d{P_{t}^{1}} = {\int}_{x_{1}}^{x_{n}} dt = x_{n} - x_{1} $$

(12a)

$$ \begin{array}{@{}rcl@{}} S_{2} \!&=&\! {\int}_{x_{1}}^{x_{n}} d{P_{t}^{2}} = {\int}_{x_{1}}^{x_{2}} \frac{y_{2} - y_{1}}{x_{2} - x_{1}} dt + {\ldots} + {\int}_{x_{i}}^{x_{i+1}} \frac{y_{i+1} - y_{i}}{x_{i+1} - x_{i}} dt \\&&+ {\ldots} +{\int}_{x_{n-1}}^{x_{n}} \frac{y_{n}-y_{n-1}}{x_{n}-x_{n-1}} \end{array} $$

(12b)

The equations for the second level signatures are provided in Table 9.

Appendix B. Classification results

Table 9 Closed-form expressions for the first and second level signature paths for landscape functions

Table 10 Persistence landscape results with support vector machine (SVM) classifier

Table 11 Persistence landscape results with logistic regression (LR) classifier

Table 12 Persistence landscape results with random forest (RF) classifier

Table 13 Persistence images results with pixel size = 0.1

Table 14 Persistence images results with pixel size = 0.05

Table 15 Template function results for H₀ diagrams

Table 16 Template function results for H₁ diagrams

Table 17 Carlsson coordinates results

Table 18 Kernel method results with LibSVM package

Table 19 Path signature method results obtained with SVM classifier