Fuzzy Logic Models

Fuzzy applications in machining process have become the focus of significant attention during the last two decades, and that is why much relevant research has been conducted. Fuzzy system theory has a wide range of applications in the machining process field and new directions are constantly given in machining process research.

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Lee et al. [59] used a thermal error model with two temperature variables using fuzzy logic modelling. However, the membership functions typically have to be manually adjusted by trial and error. The fuzzy model performs like a white-box, meaning that the model designers can explicitly understand how the model achieved its goal. However, such models that are based only on expert knowledge may suffer from a loss of accuracy due to engineering assumptions [60]. Conversely, ANNs can learn from the data provided without

preconceptions and assumptions. However, they perform as a “black-box,” which means that

there is no information regarding the method by which the goal is achieved and so the achieved optimal solution can exhibit unrealistic physical characteristics that do not extrapolate to other situations. Furthermore, the final model is not easily understandable by domain experts for interpretation purposes [61].

In order to overcome the drawbacks of traditional artificial intelligence techniques such as ANNs and fuzzy logic, more attention has been focussed on hybrid models. For instance, applying the ANN technique to optimise the parameters of a fuzzy model allows the model to learn from a given set of training samples. At the same time, the solution is mapped out into a Fuzzy Inference System (FIS) that can be evaluated by the model designer as to produce a realistic representation of the physical system. The Adaptive Neuro-Fuzzy Inference System (ANFIS) is such a neuro-fuzzy technique. It combines fuzzy logic and neural network techniques in a single system.

Wang [62] also proposed a thermal model merging Grey system model GM (1, m) and an

ANFIS. A hybrid learning method, which is a combination of both steepest descent and least squares estimator methods, was used in the learning algorithms. Experimental results indicated that the thermal error compensation model could reduce the thermal error to less than 10 µm under real cutting conditions. The author used six inputs with three fuzzy sets per input, producing a complete rule set of 729 (36) rules in order to build an ANFIS model.

Clearly, Wang’s model is practically limited to low dimensional modelling.

Eskandari et al. [63] presented a method to compensate for positional, geometric, and thermally induced errors of three-axis CNC milling machine using an offline technique. Thermal errors were modelled by three empirical methods: MRA, ANN, and ANFIS. To build their models, the experimental data was collected every 10 minutes while the machine was running for 120 minutes. The experimental data was divided into training and checking sets. Their results found that ANFIS was a more accurate modelling method in comparison

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with ANN and MRA. Their test results on a free form shape show average improvement of 41% of the uncompensated errors. However, the ANFIS model in this study is also limited to low dimensional modelling.

Construction of the ANFIS model using a data-driven approach usually requires division of the input/output data into rule patches. This can be achieved by using a number of methods such as grid partitioning or the subtractive clustering method [64]. However, one limitation of standard ANFIS is that the number of rules rises rapidly as the number of inputs increases (number of input sensors). For instance, if the number of input variables is n, and M is the partitioned fuzzy subset for each input variable, then the number of possible fuzzy rules is

Mn. As the number of variables rises, so the number of fuzzy rules increases exponentially,

increasing the load on the computer processor and increasing memory requirements. Thus, a reliable and reproducible procedure to be applied in a practical manner in ordinary workshop conditions was not proposed. It is important to note that an effective partition of the input space can decrease the number of rules and thus increase the speed in both learning and application phases. A fuzzy rule generation technique that integrates ANFIS with Fuzzy C- Means (FCM) clustering is applied in order to minimise the number of fuzzy rules. The FCM is used to systematically create the fuzzy MFs and fuzzy rule base for ANFIS. For more detail about this system, the reader is referred to section 3.3.

An accurate, robust thermal error prediction model is the most significant part of any thermal compensation system. In recent years, it has been shown that thermal errors can be predicted by empirical modelling techniques such as multiple regression analysis [45], types of artificial neural networks [56], fuzzy logic [59], an adaptive neuro-fuzzy inference system [50, 62], Grey system theory [47] and a combination of several different modelling methods [53, 57].

A common omission in the published research is discussion or scientific rigour regarding the selection of the number and location of thermal sensors. The following section explores the effect of the number and locations of sensors on the thermal model output.