Machine Tool Volumetric Error Feature Extraction and Classification Using Principal Component Analysis and K- means Methods
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1 7th International Conference on Virtual Machining Process Technology (VMPT), Hamilton, May 7-9, 2018 Machine Tool Volumetric Error Feature Extraction and Classification Using Principal Component Analysis and K- means Methods Kanglin Xing, JRR. Mayer, Sofiane Achiche Department of Mechanical Engineering, École Polytechnique (Montréal), P.O. Box 6079, Station Downtown, H3C 3A7 Montréal, QC, Canada Abstract: Volumetric errors (VE) are related to the machine tool accuracy state. The VE features affect the classification results of machine condition classifiers. Therefore, feature extraction and classification are one of the key issues to apply VE in machine tool condition monitoring. This paper presents a study that uses principal component analysis (PCA) to extract the features of VE and uses the K-means method for machine tool state classification. The VE data measured from the experimental machine tool with different states are analyzed with the above methods as a first step towards developing a condition monitoring and diagnosis solution based on VE data. The results indicate that PCA method is capable of extracting the VE feature information. These VE features can be then be efficiently classified by K-means method. The VE monitoring plan based on the two methods can be applied in machine tool condition monitoring. Keywords Machine tools; Volumetric errors; Feature extraction; Feature classification; Principal component analysis; K-means method 1. Introduction Modern manufacturing demands high machining productivity and high accuracy. Therefore, monitoring the machine tool state is a necessary part of modern manufacturing. Currently, a variety of approaches are applied for machine tool condition monitoring. They mostly monitor the machining process and mechanical structures of machine tools by signals such as the vibration, power, current, acoustic emission et al. [1, 2]. Volumetric errors (VE) are related to the machine tool mechanical structures and components such as the linear and rotary axes. It can be possibly used for monitoring the machine tool accuracy states. For the application of VE, most works are found in VE modelling, estimation[3-6]. Rarely research has been seen in machine tool condition monitoring. The key issues in the machine tool accuracy monitoring are to extract the VE features and accurately classify the machine tool accuracy states. The general feature extraction methods include Independent Component Analysis (ICA), Principle Component Analysis (PCA), Nonlinear Principal Components Analysis (NPCA). etc. [7]. PCA is an unsupervised technique. It was first proposed to decorrelate the statistical dependency between valuables in multivariate statistical data [8]. Since then, it has been widely applied in the areas such as statistical analysis, process monitoring and diagnosis and pattern recognition [8]. PCA method is a simple nonparametric method which can extract the most relevant information from a set of redundant or noisy data. These most relevant information forms some new variables, the principal components, and explained the maximum amount of variability of the original data. Based on the successes of cluster generation, clustering techniques can be classified as hierarchical Kanglin Xing 2017
2 clustering, partitional clustering, graph theory based clustering, fuzzy based clustering or neural networks based clustering, etc. [9]. As a squared error-based clustering method, the K-means algorithm can be simply implemented in solving many practical problems [9-11]. In this research, we explore how to apply the PCA method to extract the VE features and use the K-means methods to classify the machine tool states indicated by VE features, the results will be later used for developing a VE based condition monitoring solution. The paper begins by presenting state-of-the-art in VE, PCA and Clustering. The VE feature extraction plan follows VE features clustering. Then, testing data is introduced, followed by the experimental results and their analysis. Finally, the conclusions are presented. 2. VE and its monitoring plan VE is defined as the relative Euclidian error vector between the tool frame and workpiece frame in 3D space [5] (Fig. 1). It can be written as VE i = [VEx i, VEy i, VEz i ] where i stands for the i th of VE measurement positions in one test. A vector similarity measure -module is used to calculate the norm of VE. Therefore, the basic VE dataset of one measurement can be written as VEM i 1 = [ VE 1, VE 2, VE 3,, VE i ]. For the periodical monitoring VE data, it can be expressed as VEM i j where j represents the VE testing times. Fig. 1. VE of the HUT-40 5-Axis machine tool, it contains three linear axes (X, Y, Z) and two rotary axes (B, C). The typology structure of this machine tool is WCBXFZYST. Where W means workpieces, T means tools, and F means the foundation frame of machine tool The functional information flow of the VE monitoring system is shown in Fig. 2. During the machine tool maintenance period, accuracy measurement devices/methods will be run to acquire the VE data. Then PCA method will extract the new VE feature from the original VE data. The new VE features will be classified by K-means methods to check the states of VE data. Then, the change of the states of the machine tool can be revealed for the machine tool operators. VE measurement VE feature extraction VE feature classification Fig. 2. Flowchart of VE monitoring system 3. VE feature extraction To simplify the processing of VE data, PCA is used to extract the new features from the VEM i j (i means the VE measurement positions in one VE measurement test, and j means the total VE measurement times) by dimensionality reduction. In this paper, we will not discuss the Kanglin Xing
3 mathematical details of PCA. The complete details on PCA data treatment as well as the application of PCA can be found in [8]. The general steps of PCA in VE data feature extraction are as follows: 1. VE data preparation. 2. Subtract the mean from each corresponding input feature VEM i j to create a new normalized matrix. 3. Calculate the covariance matrix of the new normalized matrix. 4. Calculate the eigen values and eigen vectors of the covariance matrix. 5. Choose the components by considering the cumulative percent variance (CPV) which denotes the approximation precision of the new largest eigenvectors. 6. Calculate the final projected data set which represents the modelled variation of VEM i j based on first N components. The initial data set VEM i j is finally projected on to a new structure with new sets of data matrix PVEM i N. And B i N is the matrices of N retained eigenvectors. PVEM j N = VEM j i B i N (1) It is worth noting that CPV ( 90%) was set to select the 1 st and N th principal components. And the VE feature extraction is automatically processed with PCA coded using Matlab. 4. VE features classification K-means is a vector quantization method and is a very popular for cluster analysis. The main aim of K-means clustering in VE feature classification is to classify N different VE features into K clusters where each VE feature observation belongs to the cluster. The observation which belongs to the cluster is assumed to have the nearest mean which generally serves as a prototype of the cluster [12]. Then, VE data measured from the machine tool with the same condition can be grouped together. Otherwise, they will be put in a different cluster. This would allow the development of a signature file of machine tool behavior based on VE. The VE feature classification is automatically processed using K-means. K-means contains two main parts: the first part is to select K centers randomly, where the value of K needs to be fixed in advance. The next part is to assign each VE feature data object to the cluster with the nearest center. The steps for K-means classification are: Step 1: Prepare the VE feature data PVEM j N = [ PVEM j 1, PVEM j 2, PVEM j N ] Step 2: Randomly select K cluster center setups. Step 3: Calculate the Euclidean Distance between each data object PVEM a N (1 a j) and all k cluster centers Cn(1 n k) and assign data object to the nearest cluster. This process is expressed mathematically as follows: j KD = min PVEM a N C n b=1 2 (n (1, 2, k)) (2) Step 4: Update the cluster center at periodic intervals. Step 5: Repeat the steps 2 to 4 until there is no change in the sum value of the total squared errors for each cluster center. 5. Testing data and results 5.1. Testing data from real machine tool measurement We chose the SAMBA method to estimate the VE in this research because of its advantages in simple installation and maintenance, automated data acquisition and processing [13]. The SAMBA test is carried out on the HU40-T five-axis machine tool using a Renishaw probe while performing the indexation set (4 balls and 28 indexations of the B and C axes) (Fig. 3). The measured master ball artefacts coordinate information are used for the SAMBA mathematical model. The machine tool has been periodically tested twice per week (76 in total). The estimated Kanglin Xing
4 VE result (76 times) with the machine tool in five different states: normal state 1, fault states (C axis encoder fault, uncalibrated C axis encoder fault and pallet location fault) 2, 3 and 4, and another normal state 5 different from the state 1, were used in this research. Fig. 3. SAMBA measurement 5.2. VE feature extraction Fig. 4. Contributions of principal components The VE data measured from the machine tool are firstly processed with the Module measure. Then, all the data will be processed with the PCA program. Fig. 4 illustrates the contribution of the new principal components of the VE feature data. The first and second component contain 78.5% and 13.7% of VE variation respectively, which sums up to a total of 92.2%. This means the subspace composed of the two components includes enough variation information of the original features. Therefore, the two components are extracted as the new features of VE. After the PCA processing, the dimension of the original VE data is decreased from 109*76 to 2*76. The PCA data processing method obviously reduced the difficulty of VE data processing. Fig. 5. Variation tendency of new VE features indicating machine tool with five states Although the original VE data contributes to all the four new principal components, they are not all efficient for machine tool condition recognition, whereas the first and second features can Kanglin Xing
5 identify the different states of the machine tool (Fig. 5). The remaining two principal components are unable to separate the transition of machine tool states, and their curves do not have a similar change tendency. Therefore, the adding of the two extra components would probably add unnecessary noise to the machine tool state recognition VE feature classification Using the new extracted VE features, the first and the second principal components are processed with the K-means method for feature classification. Fig. 6 illustrates the classification results from using K-means and PCA methods. The VE for this research are measured from machine tool experienced five states: normal state-1, fault states-2, 3 and 4, and another normal state-5. The five different states are not only classified by K-means methods but are also classified by PCA method. For the propose of comparison for the PCA-results, different colors have been manually added to the components according to the VE testing sequence. PCA-results reveal that the VE data belonging to the same machine tool condition can be classified in one single cluster. K-means results are generated automatically without revision. For the machine tool fault states 2, 3 and 4 can be well categorized by K-means and PCA methods. However, for the machine tool normal states 1 and 5, they can only be classified by K-means methods although some points are not classified correctly (Fig.6, see the points circled). PCA method does not differenciate between the normal states 1 and 5 because they are very close. To see the differences between these two similar states, K-means method can achieve a better classification by adjusting the original setup K value (Part 4, step 2). Therefore, for the machine tool states classification, PCA method can be firstly used for the rough classification of VE data. The obtained cluster number can then be used as a reference for K value setup. Then, by adjusting the K value, more precise classification can be achieved. Fig. 6. Classification results of new VE features with K-means and PCA methods 6. Conclusions This paper mainly presents how to use principal component analysis (PCA) to extract the features of VE and use the K-means method to classify the machine tool states. The testing results indicate that the two proposed methods are effective in their applications, in addition, PCA can Kanglin Xing
6 also classify the machine tool states. However, compared to K-means method, it still lacks a deeper classification capability for optimal machine tool similar states recognition. Therefore, the combining of PCA and K-means methods can improve the accuracy of the machine tools states classification. By calibration, the plan based on PCA and K-means is hoping to be applied to automatic VE changing point detection. Based on the current work, the future step of this research is developing a machine tool accuracy condition diagnosis system. 7. Acknowledgements The authors thank the technicians Guy Gironne and Vincent Mayer for conducting the experimental part of this work. This research presented in this paper was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) under the CANRIMT Strategic Research Network Grant NETGP and China Scholarship Council (CSC). 8. References 1. Sofiane Achiche, M.B., Luc Baron, Krzysztof Jemielniak, Tool wear monitoring using geneticallygenerated fuzzy knowledge bases. Engineering Applications of Artificial Intelligence,, (3-4): p Q. Ren, M. Balazinski, K. Jemielniak, K.L. Baron, S. Achiche., Experimental and fuzzy modelling analysis on dynamic cutting force in micro milling. Soft Computing, (8): p Zhang, Y., et al., Volumetric error modeling and compensation considering thermal effect on fiveaxis machine tools. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, (5): p Rahman, M.M.a.J.R.R.M., Five axis machine tool volumetric error prediction through an indirect estimation of intra- and inter-axis error parameters by probing facets on a scale enriched uncalibrated indigenous artefact. Precision Engineering, : p Vahidi Pashaki, P. and M. Pouya, Volumetric error compensation in five-axis cnc machining centre through kinematics modeling of geometric error. Advances in Science and Technology Research Journal, (30): p Givi, M. and J.R.R. Mayer, Volumetric error formulation and mismatch test for five-axis CNC machine compensation using differential kinematics and ephemeral G-code. The International Journal of Advanced Manufacturing Technology, (9-12): p Khalid, S., T. Khalil, and S. Nasreen. A survey of feature selection and feature extraction techniques in machine learning. in 2014 Science and Information Conference Jackson, J.E., A User s Guide to Principal Components. Vol , New York: John Wiley & Sons. 9. Rui, X. and D. Wunsch, Survey of clustering algorithms. IEEE Transactions on Neural Networks, (3): p Schlechtingen, M., I.F. Santos, and S. Achiche, Using Data-Mining Approaches for Wind Turbine Power Curve Monitoring: A Comparative Study. IEEE Transactions on Sustainable Energy, (3): p Raouafi, S., et al., Classification of upper limb disability levels of children with spastic unilateral cerebral palsy using K-means algorithm. Medical & Biological Engineering & Computing, (1): p Su, M.C. and C.H. Chou, A Modified Version of the K-Means Algorithm with a Distance Based on Cluster Symmetry. IEEE Trans. Pattern Anal. Mach. Intell, : p Mayer, J.R.R., Five-axis machine tool calibration by probing a scale enriched reconfigurable uncalibrated master balls artefact. CIRP Annals - Manufacturing Technology, (1): p Kanglin Xing
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