Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 8281490, 10 pages

http://dx.doi.org/10.1155/2016/8281490

## Thermal Error Modelling of the Spindle Using Neurofuzzy Systems

^{1}School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China^{2}College of Mechanical and Electrical Engineering, Shihezi University, Shihezi 832003, China^{3}Laser Institute of Shandong Academy of Sciences, Jinan 250000, China

Received 10 November 2015; Revised 11 February 2016; Accepted 21 February 2016

Academic Editor: Mohammed Nouari

Copyright © 2016 Jingan Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a new combined model to predict the spindle deformation, which combines the grey models and the ANFIS (adaptive neurofuzzy inference system) model. The grey models are used to preprocess the original data, and the ANFIS model is used to adjust the combined model. The outputs of the grey models are used as the inputs of the ANFIS model to train the model. To evaluate the performance of the combined model, an experiment is implemented. Three Pt100 thermal resistances are used to monitor the spindle temperature and an inductive current sensor is used to obtain the spindle deformation. The experimental results display that the combined model can better predict the spindle deformation compared to BP network, and it can greatly improve the performance of the spindle.

#### 1. Introduction

Accuracy of machined work pieces is one of the most critical considerations for any manufacturer. In all components of machine tools, the spindle is the most important component because it provides cutting power and is part of the force chain between the machine tool structure and the tool. It directly affects the accuracy of work pieces and is one of the main error sources in terms of its contribution to the total heat generation and the resulting deformations [1]. Hence, the performance of the spindles directly determines the machine tools entire performance [2].

The accuracy of machine tools depends on positioning errors. In the overall position errors, thermal errors caused by internal heat sources and environment are up to 70% [3]. Thermal error is a time-varying and nonlinear procedure induced by nonuniform temperature variation. The interactions between the thermal expansion of components, heat sources, and heat conduction produce complex thermal behavior. For machine tools, constructing a precise structure model is extremely difficult, very costly, and time-consuming, and much easier methods are to use error compensation [4]. With effective compensation, using medium precise machine tools can manufacture work pieces with higher accuracy [5]. Thermal error compensation has become a cost-effective method to improve accuracy of machine tools, especially with the increasing demand for machining accuracy in recent years.

Since the two keynote papers, the first about thermal effects from [6] and the second about error reduction and compensation of machine tools from [1], a lot of research in this area has been performed. For the error caused by the heat deformation, compensation methods are divided into direct and indirect compensation. Direct compensation directly measures the drift displacements between the tool and the work piece to compensate positioning errors. Indirect compensation uses mathematical or physical models to find the relationship between auxiliary values (such as temperature variables) and thermal deformation. The output of the models is used to compensate positioning errors. In many situations, direct compensation is often quite difficult because the measurement of the drift is not always possible during the machining process. Hence, indirect compensation is more convenient and easier. Researchers have developed many indirect compensation methods, such as finite element analysis [7–9], regression analysis [10–12], fuzzy logic [13, 14], neural networks [15–17], and the combination of two or three methods [18–20].

Currently, most current research focuses on using ANNs (artificial neural networks) to build error compensation models based on temperature variables. Compared with other models, ANNs has the advantages of parallel processing, information distribution saving, and self-learning ability. In recent years, different types of ANNs have been developed, including radial basis function (RBF) network [15, 21], feed-forward neural networks [22], backpropagation (BP) network [23], grey neural network [19], Elman network [24], integrated recurrent neural network [25], and cerebellar model articulation controller (CMAC) neural network [26]. However, these neural-modelling methods have poor generalization capability and are sensitive for external noise. For the random initialized weights, the learning course may trap in local minima, and some neurons may give rise to saturation. So traditional ANNs is not fit for modelling the time-varying and nonlinear procedure of thermal error.

Adaptive neurofuzzy inference system (ANFIS) used in this paper is a neurofuzzy approach. It combines the fuzzy logic qualitative characteristics and neural network adaptive capabilities. Therefore, it is more flexible on structure, and it can more effectively approximate a highly nonlinear surface than traditional ANNs. As a neurofuzzy modelling method, it has been widely used in different fields, such as prediction [27, 28], knowledge discovery [29], control system [30, 31], and spattern recognition [32].

In this paper, we propose a new combined model to predict the spindle deformation, which combines the grey model and the ANFIS model. The grey models are used to preprocess the original data, and the ANFIS model is used to adjust the combined model. To evaluate the performance of the model, the experiment for the spindle is implemented. The results show that the combined model has high prediction accuracy, and it has better performance than BP networks.

The paper is organized as follows. Section 2 presents our proposed model combining the grey models and the ANFIS model. Section 3 describes the experimental setup. Section 4 describes the experimental results and performs comparisons of the combined model and BP network. Finally, conclusions are presented in Section 5.

#### 2. Building the Predictive Model

##### 2.1. Data Preprocessing

In order to reduce the randomness of the original data and the influence of unpredictable noises, we use the grey model to preprocess the original data, namely, the measured data of the spindle. Grey system theory is a kind of systematic and scientific theory developed originally by Deng [33] in the 1980s.

Based on the original data of the spindle, we establish the grey model GM, which is a first-order grey model with variables. The following steps are performed.

*Step 1. *Create the original data sequence containing the temperature and thermal deformation of the spindle and it is expressed as follows:where is the thermal deformation data sequence, is the temperature data sequence, and is the sample size of the data.

*Step 2. *Using the accumulated generating operation (AGO), we convert chaotic series into monotonously increasing series , and it is given bywhere is calculated by

*Step 3. *With the following first-level mean generating operation (MGO), we create the background series from , and it is expressed as follows:

*Step 4. *Establish the grey differential equation of the grey model GM, and it is expressed as follows:where the parameters can be obtained by using the least-square method as follows:where

*Step 5. *Set up the grey model GM as follows:where is the prediction value at a time . Using the first-level inverse accumulated generating operation (IAGO), we can obtain the output of the grey model GM:In the data preprocessing, we use the data sequence where , to predict , namely, the thermal deformation of the spindle. We employ three groups of grey models, which have different length of the data sequence. The outputs of the grey models are used as the inputs of our proposed ANFIS model, together with the measured thermal deformation used as the output, to train the ANFIS model.

##### 2.2. Model Adjustment Using ANFIS

For getting better predictive effect, we use ANFIS to perform the adjustment of the combined model. ANFIS is a fuzzy inference system (FIS) implemented as a neural network, firstly proposed by Jang [34] in 1993, and then has been widely used [35–37]. It is a five layered feed-forward neural network structure and uses fuzzy reasoning and neural network learning algorithms to map inputs into an output. The ANFIS architecture used in this paper is shown in Figure 1. It uses the first-order Sugeno fuzzy model and has three inputs linked with three membership functions (MFs), eight rules, and one output.