Data-Driven Models

The data-driven models are behavioural models that are based on historical data to predict the performance of a given system, in this case the distortion of the machine. Contrary to the

numerical models, they are not based on explicit physical equation definitions but on experimental database which is capable of reflecting the relationship between inputs and

outputs. Data-driven techniques for thermal error modelling can be divided into two categories: statistical techniques such as regression methods, linear polynomial models, etc.,

and artificial intelligence techniques such as Artificial Neural Networks (ANNs), fuzzy

systems, etc.

2.1.2.1 Statistical Models

Linear regression is the simplest method to correlate measured temperatures with resulting displacement. A Least Squares (LS) approach is used to obtain the coefficients that determine the relationship between inputs and output without using any physical equation. Although this method can provide reasonable results for a given machine test regime, the thermal displacement usually changes with variation in the machining process and the environment, which introduces an error into the model [43]. The linear regression model is also time- consuming and labour intensive to design.

Hardwick [44] established a Multiple Regression Analysis (MRA) model of a vertical machining centre and a horizontal machining centre. The author highlighted the complexity of the interrelation between the different heat sources and the different mechanisms of response. Test results showed that the thermal error could be reduced to less than 7 µm, representing an improvement in accuracy of almost 7 times.

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Chen et al. [45] used an MRA model for thermal error compensation of a horizontal machining centre. With their experimental results, the thermal error was reduced from 196 to 8 µm. Yan et al. [46] also used the MRA model to form an error synthesis model, which merges both the thermal and geometric errors of a lathe. With their experimental results, the error could be reduced from 60 to 14 µm.

Grey model is another statistical approach that has been used for thermal error compensation system [47]. Grey system theory is a method introduced by Deng in early 1980s [48] with the intention to study the Grey systems by using mathematical methods with poor information

and small datasets. In Grey system theory, GM (h, N) denotes a Grey model, where h is the

order of difference equation and N is the number of variables. The GM (h, N) model can be

used to describe the relationship between the influencing sequence factors and the major sequence factor of a system. Furthermore, weights of each factor represent their importance to the major sequence factor of the system. Its most significant advantage is that it needs only a small amount of experimental data for accurate prediction, and the requirement for the data distribution is also low [49]. For more detail about Grey models, the reader is referred to section 3.4.

Wang et al. [47] proposed a systematic methodology for the thermal error compensation of a machine tool. The thermal distortion was modelled using a Grey model based on Grey system theory to predict the thermal errors with only 30 minutes of measured data. Unfortunately, their model was lacking in the ability of self-learning, self-adaption, self-organisation, and taking feedback correction into consideration. Therefore, their model obtained under one particular operating condition is still not robust under other operation conditions.

In order to overcome the drawbacks of statistical models, more attention has subsequently been given to the artificial intelligence techniques such as artificial neural networks.

2.1.2.2 Artificial Intelligence (AI) models

Artificial neural network as a form of AI is a data-driven approach. It is designed in a way

that mimics the behaviour of biological neural network. A typical artificial neural network has an input layer, one or more hidden layers, and an output layer [50]. The neurons in the hidden layer, which are connected to the neurons in the input and output layers by adaptable weights, enable the ANN to compute complex associations between the input and output variables [51]. The inputs of each neuron in the hidden and output layers are summed and the

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resulting summation is processed by an activation function [51]. Training the model is the process of determining the adjustable weights and it is similar to the process of determining the coefficients of a regression model by least squares approach. The weights are initially selected randomly and an optimisation algorithm is then used to find the weights that minimise the differences between the model-calculated and the experimental outputs [52]. Across the whole modelling procedure, no physical equation is used.

To find the relationship between inputs and outputs of a complex thermal behaviour, ANN techniques have drawn more attention rather than statistical models, and produce results without requiring a detailed mechanistic description of the phenomena that is governing the system. There are different ANN architectures to building thermal models, Back-Propagation (BP) artificial neural network modelling has proved to be a suitable nonlinear modelling method [50, 53].

One of the major advantages of ANNs is efficient handling of highly non-linear relationships in data. In recent years, it has been shown that thermal errors can be successfully predicted by artificial neural networks [50, 54, 55].

Chen et al. [56] proposed an ANN model structured with 15 nodes in the input layer, 15 nodes in the hidden layer, and six nodes in the output layer in order to drive a thermal error compensation of the spindle and lead-screws of a vertical machining centre. The ANN model was trained with 540 training data pairs and tested with a new cutting condition, which was not included within the training pairs. Test results showed that the thermal errors could be reduced from 40 to 5 µm after applying the compensation model, but no justification for the number of nodes or length of training data was provided, so the scalability of the method cannot be assessed. Moreover, the compensation system must be flexible to extension to other physical inputs, meaning that alternative variables can be deployed with minimal effort. Wang [57] used a neural network trained by a Hierarchy-Genetic-Algorithm (HGA) in order to map the temperature variation against the thermal distortion of the machine tool. Experimental results indicated that the thermal error compensation model could reduce the thermal error to less than 10 µm under real cutting conditions.

Zhu et al. [58] presented a clustering approach based on correlation coefficient to pick out three optimal temperature variables. The output-hidden feedback Elman neural network is adopted to establish a model for the prediction of thermal errors on a CNC machine tool. The verification experiment shows that the combination of clustering and neural network model is

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a good way for thermal error compensation. However, a small number of temperature sensors may lead to poor prediction accuracy under different operation conditions.

ANN models may be successfully applied in thermal error compensation systems and can capture effectively the non-linear relationships existing between variables in nonlinear systems such as thermal modelling. However, ANN models are considered as black-box models, because the mathematical relations in these models are unknown to the designer and have no physical meaning. Moreover, it is worth pointing out that these models need proper training to work effectively, and that their performance is limited by their design parameters. Therefore, using ANN involves a moderately tedious trial and error effort for obtaining the network structure, especially involving the hidden layer neurons, number of hidden layers, transfer function, training algorithm and learning rate parameter.

The purpose of this section was to give an overview about thermal modelling and to show the important role of data-driven models in this field of thermal error compensation. It was shown that ANN models are general black-box modelling tools, which have many attractive features: they are constructed without any physical laws but only a set of input-output data for training procedure. The training data has to cover the whole expected range of the operation conditions. However, the process of obtaining such data can take several hours for internal heating tests and several days or more for the environmental test [3].

An important implication obtained from the above mentioned models is that they have been proven for a single error source but not a whole (complete) machine structure. Furthermore, most of them have used only one approach, which usually cannot result in satisfactory thermal model. Combining different AI techniques, however can join the advantages of the different methods, can utilise different representations of knowledge, and can help to understand the result obtained. The latter is especially important in ANN modelling, because ANN models cannot give explanation of the system, and without explanation, the lack of physical meaning may reduce the acceptance of the ANN models even if their result are in good agreement with the experimental data.