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ABSTRACT This paper presents a new approach to sensor based feature evaluation and selection for modelling purposes using a Self-Organizing Map. Self-Organizing Maps perform classification in a non-supervised fashion performing vector quantization and therefore place similar vectors close together in the two dimensional output space. The unsupervised process leads to the self organization of modelling with no previous knowledge of what is being modelled and therefore it does not model a predetermined environment. Taking the above into account feature selection was performed by analysing the contributions of different sensor based features, carrying large quantities of noise, towards tool wear classification. It was found that some of the features, not previously evaluated and justified, have a strong contribution towards tool wear classification. It is demonstrated that the use of the self-organizing map can be used to quantitatively evaluate the contribution of features towards neural network modelling of systems in the presence of noisy data. 1. Introduction Since the late 1990s we have witnessed a change from the old practice of changing tools automatically, to the feasibility of instituting tool change procedures based on monitoring the amount of wear on the cutting tool-edges through the implementation of adaptive tool inspection mechanisms. Literature reports numerous proposals for the architecture of condition monitoring systems for online supervision and control. Regardless of the numerous contributions towards the scientific advance in this field none of the architectures have gained sufficient acceptance or otherwise proved to be feasible towards most machining processes/conditions. One important strategy to support this goal is sensor-based, real-time control of key characteristics of both machines and products, throughout the manufacturing process. The development of such systems takes into account the traditional ability of the operator to determine the condition of the tool based on his/her experience and senses, e.g. vision and hearing. Thus, the importance of sensor-based information and its deterministic nature is fundamental towards the delivery of reliable condition monitoring systems as well as might contribute towards the development of new approaches towards unmanned machining. The literature reports innumerous examples pertaining to the study of sensor feasibility regarding tool condition monitoring (****) and from this research a large variety of sensors have been proposed (***). Previous work on the relationship between audible emissions and tool wear has established that audible emissions are capable of indicating the extent of the cutting edge wear, Weller et al.[1]. McNulty et al. [2] have also highlighted the use of noise spectra for tool life evaluation applied to several cutting processes and have found significant changes in certain frequency bands that appear to be characteristic of wear in certain cutting processes. Lee [3] found that, during turning the machine noise exhibited a wear related change of sound pressure level (SPL) at certain frequencies (4 – 6 kHz) for several materials. A drop in the SPL before the tertiary zone (third and last stage of wear) was suggested as an end of tool-life predictor. Experiments carried out by Ya et al. [4] using two different types of turning tool showed that both the tool angle and the cutting speed exerted no great influence on the average cutting noise. Vibration has also been used to recognize the wear state of a tool whilst turning [5] and the main advantage of this method is its ease of application. Wear monitoring of cutting tools has been performed using many different sensing techniques. These techniques include; temperature, motor current, acoustic emission (AE), audible emissions, vibration and force, Dimla E. Dimla Snr. [8] and Kunpeng et al. [9]. Some of these have been successfully applied under laboratory conditions although industrial applications have been rather unsuccessful. These are just but a few of the original contributions that support the interest and sustain the importance of unmanned production. Despite the enormous amount of work developed so far it is recognized that forecasting in complex systems that are poorly understood, noisy and often non-linear can be practically impossible when based on the traditional model predictive algorithms, Parlos et al. [6] and Li [7]. Consequently engineers tend to rely on system identification techniques to establish process models. Monitoring tool condition encompasses the acquisition of noisy data that has to be filtered, selected and synthesized into carefully selected indicators. Clearly, the quality of the sensor information leads to promising results concerning the detection of the state of wear in idealized conditions but much work has to be performed in information processing and decision making in order to correctly classify the tool wear state from the available sensors under different and vast cutting conditions (****). The common factor in all these research topics is the development of an effective Tool Condition Monitoring System. A reliable system should integrate different information sources and must be based on reliable sources so that a robust system could be built. A clear understanding of the sensed data and its dynamical behavior is fundamental for a proper extraction and understanding of condition monitoring indicators. Nonlinearities arising out of different sources such as mid-plane stretching and electrostatic force lead to a rich nonlinear dynamics in the time response of these systems which should be investi- gated for appropriate design and fabrication of them. Motivated by this need, present study is devoted to analyzing the nonlinear dynamics and chaotic behavior of nano resonators with electrostatic forces on both sides. Based on the potential function and phase portrait of the unperturbed system, the resonator dynamics is categorized to four physical situa- tions and it is shown that the system undergoes homoclinic and heteroclinic orbits which are responsible for the appearance of chaos in the resonator response. Bifurcation diagram of nano resonator is plotted by variation of applied AC actuation voltage and it is shown that the system possess rich dynamic behavior such as periodic doubling, quasi-periodic, bifurcation and chaotic motion which are classified and studied in more details by plotting time response and phase plane of the each category. Although a newcomer, chaos theory has proven to find a place in different fields of research such as energy (Karatasou and Santamouris [10]), health (Oliveira et. al, 2011), computing (Hu et. Al , 2012) and hydrology (Khatibi et. al, 2012), among others. Given the noisy characteristic of sensor based information and the challenging task of modelling the cutting process most often Tool wear monitoring is performed using artificial intelligence techniques such as neural networks ([11]; [12]). However, if model-based analysis can be criticized for its simplistic models, then nonlinear time-series analysis can be criticized for its assumed generality. Although it can be used for a wide variety of applications, it contains no physics. It is dependent Phil. Trans. R. Soc. Lond. A (2001). Nonlinear models for complex dynamics in cutting materials on the data alone. Thus the results may be sensitive to the signal-to-noise ratio of the source measurement, signal ­ filtering, the time delay of the sampling, the number of data points in the sampling and whether the sensor captures the essential dynamics of the process. 2. Determinism and Chaos in Cutting Operations A deterministic system can evolve in a way which, in the long term, is unpredictable. The analysis of this kind of evolution is the objective of the theory of chaos. Applications of this theory include physics, bioinformatics, biomedicine, meteorology, chemistry, sociology, astrophysics, engineering, economy. An excellent review of the history of the concepts underlying the- ory of chaos, from the 17th century to the last decade, has been given by Christophe Letellier in his book ‘‘Chaos in nature’’ [1]. Though Chaos is a phenomenon of disorderly-looking long-term evolution occurring in a deterministic nonlinear system (Williams, 1997) whereby its behavior is sensitively dependent on initial conditions, topological transitionanddensity of periodic points (Devaney, 1989). Chaos is used in the literature to refer to the dynamics of complex and random-like situations arising from simple nonlinear deterministic systems sensitive to initial conditions (e.g. Lorenz [13]). These properties arise on the cutting process where the systems dynamics is governed by a limited number of system variables, despite their complexity and random-looking behaviour. Early studies such as the one of Grabec [14][15] proposed a nonlinear model for cutting where the normalized force is a function of the width of cut. This model is mathematically represented by a set of two second-order coupled nonlinear differential equations which are inherently unstable and not well-enough conditioned (Carnahan et al. [16]). Several other studies suggest the deterministic behaviour of cutting in the turning process. One of the fundamental questions regarding the physics of cutting solid materials is the nature and origin of low-level vibrations in so-called normal or good machining – cutting below the chatter threshold. Below this threshold, linear models predict no self-excited motion. Yet when cutting tools are instrumented, one can see random-like bursts of oscillations with a centre frequency near the tool natural frequency. Work by Johnson (1996) has shown that these vibrations are significantly above any machine noise in a lathe-turning operation (Bukkapatnam [17]; Moon & Abarbanel [18]). Typically there are two ways, i.e., graphical methods and quantifiers, to identify chaos or assess if a system is stable, periodic, quasi-periodic or chaotic. Graphs and plots are visually convenient in showing data trends and patterns. Time series plots, phase space plots, return maps and power spectrum can be used to study sys- tem behavior (Hilborn, 1994; Sprott & Rowlands, 1995). However, graphical methods are not the most accurate way to evaluate chaos. Well-defined quantifiers can be calculated to evaluate if a system is stable, periodic, quasi-periodic or chaotic. Lyapunov exponent (LE), entropy, fractal dimension, capacity dimension, and correlation dimension are some of the widely used quantifiers (Hilborn, 1994; Sprott & Rowlands, 1995). In this study, we use LE to quantify the chaos dynamics of the supply chain system as the principal quantifier because it is a proven measure for determining and classifying nonlinear system behavior (Wolf, Swift, Swinney, & Vastano, 1985). Using LE also makes our comparative analysis with the previous study (Hwarng & Xie, 2008) easier. Some real deterministic systems may display chaotic features by undergoing a loss of temporal correlation in response to small perturbations, particularly in initial conditions, but some may not. Chaotic behaviours are a reflection of internal behaviours in the time history of one (or more) of system variables, normally referred to as time series, which may therefore bear external signals for indicating their behaviours. As a way of avoiding spurious results, it is customary to employ many different techniques. This paper uses five nonlinear dynamic methods, each of which sheds light on a different aspect(s) of the system that can be identified through the concept of chaos. These are categorised as follows: (1) reconstruction of attractors in phase space: this paper uses Average Mutual Information (AMI) to quantify the delay time dimension; (2) determination of dimensionality of the trajectories: this paper uses False Nearest Neighbour (FNN) algorithm and correlation dimension method to quantify the embedding dimension; (3) and identification of convergence/divergence, predictability and prediction: this paper uses the Lyapunov exponent method and local approximation prediction method for these. The analysis provided here assumes that the underlying mechanisms of sensed signals are deterministic and chaotic, as presented by several authors such as Abarbanel ????, ???, ???, These experiments and others (Bukkapatnam et al . 1995a; b) suggest that normal cutting operations may be naturally chaotic. Further, it is also analysed and discussed the relationship between tool condition and system’s determinism and the viability of forecasting under such complex behaviour. 3. Recurrence Plots and Chaos Detection applied to Sensed Data Given the complexity of the underlying system dynamic’s we rely on time-series to perform an evaluation of the deterministic behaviour of the cutting process. Tool condition monitoring implies data collection from different sources that assume a temporal sequence format providing the basis for diagnose, i.e. x_i, where i represents the sample sequence number that finds correspondence in a time scale according to the sampling period. The Taken´s theorem (Takens [19]) states that the dynamics of a time series is fully captured in an m dimensional phase space which is as least the dimension of the original attractor.  The reconstructed phase space can be built from the time series using the delay method introduced by Fraser and Swinney [20], as follows:    (1) where  is the time delay and m the embedding dimension. Determining appropriate parameters is not trivial and includes several methods such as Average Mutual Information AMI (Fraser and Swinney [20]), Autocorrelation Function ACF (e.g. Holzuss and Mayer-Kress , 1986) or Correlation Integral CI (e.g. Leibert and Shuster, 1989). From these methods the Average Mutual Information is considered the best since it reflect non-linear properties and thus not require large amounts of data, otherwise required by CI and ACF. The AMI approach sets the time delay as the first minimum of the Average Mutual Information, a method suggested by Fraser and Swinney (1986). The AMI is given by     (2) Where p_i is the probability of finding a time series value in the i-th interval, and p_ij() is the joint probability that an observation falls into the i-th interval and the observation time  later falls into the j-th interval. To determine the embedding dimension m we use the false nearest neighbour method, Kennel et. al (1992). The method relies on the assumption that an attractor of a deterministic system folds and unfolds smoothly with no sudden irregularities in its structure. Two points that are close in the reconstructed embedding space have to stay sufficiently close also during forward iteration. If this criterion is met, then under some sufficiently short forward iteration, originally proposed to equal the embedding delay, the distance between two points y_i and y_t of the reconstructed attractor, which are initially only a small distance apart, cannot grow further then a given constant.     (3) If R_i is larger than a given threshold then y_i is marked as having a false nearest neighbour. Equation (3) has to be applied for the whole time series and for various values of m until the fraction of points for which R_i is bigger than the threshold is negligible. Lyapunov exponents are a quantitative measure for distinguishing among the various types of orbits based upon their sensitive dependence on the initial conditions, and are used to determine the stability of any steady-state behaviour, including chaotic solutions. The reason why chaotic systems show aperiodic dynamics is that phase space trajectories that have nearly identical initial states will separate from each other at an exponentially increasing rate captured by the so-called Lyapunov exponent (Linsay, 1981;  Fraser and Swinney, 1986). This is defined as follows. Consider two (usually the nearest) neighbouring points in phase space at time 0 and at time t , distances of the points in the i-th direction being ||δxi (0)|| and ||δxi (t)||, respectively. The Lyapunov exponent is then defined by the average growth rate λ_i of the initial distance.    (4) The most important observation is that the largest Lyapunov exponent denoted as 1uniquely determines whether the system is chaotic or not. Thus, for our purposes it suffices to constrain the analysis solely to the largest Lyapunov exponent. We describe an algorithm developed by Wolf et al [28], which implements the theory in a very simple and direct fashion.    (5) The first step of the algorithm consists of finding the nearest neighbour of the initial point y_k. Let L(t_k) denote the Euclidean distance between them. Next, we have to iterate both points forward for a fixed evolution time, which should be of the same order of magnitude as the embedding delay τ, and denote the final distance between the evolved points as L'(t_k-1). Recurrence plots have been first used to unveil hidden patterns and non-stationarity in dynamical systems, Eckman et. Al (1987). Recurrent plots are very appealing for finding hidden correlations on complex dynamical systems also because they do not necessarily rely on stationary data. A recurrence plot is a two dimensional (i,j) representation of the dynamical system where each point represents the distance (gray levels) between two points (y_i and y_j) in the reconstructed attractor. The interpretation of recurrence plots is fairly simple: if the signal being analysed has a random structure the plot will present itself with a uniform distribution with no evident pattern showing. If the underlying signal gives rise to a well defined set of patterns then it should have its origins in a deterministic signal. The downside of this approach is that it can only provide a qualitative assessment of the underlying signal. Homogenous recurrent plots demonstrate the stationarity nature of the underlying signal and texture of the plot gives an indication if the system is deterministic or otherwise stochastic. 4. Experimental Work Taking into consideration previous research (e.g. Jiang et al., 1987; Kunpeng et al., 2009) vibration has been chosen in this work as a secondary source of information because of the correlation between machine tool vibration and tool wear that have been demonstrated successfully in the laboratory. Based on the above considerations experimental background work was conducted on the turning process to collect tool wear data. In this work a set of tool wear cutting data was acquired by machining a block of mild steel under realistic production conditions that consisted of a cutting speed of 350 m/min, a feed rate of 0.25 rev/min and a depth of cut of 1 mm, with a coated cemented carbide tip. The set of sensors used were: an accelerometer for measuring vertical vibration, a microphone for recording sound emission, a strain gauged tool holder for force measurement and a meter for the spindle current of the CNC machine. The turning operation was carried out on an MT 50 CNC Slant Bed Turning Centre. The analogue signals were sampled at 20 kHz with tool wear and sensor data being acquired at intervals of 2 min, taking into account an expected tool life, for each insert, with a typical value of 15 min. Sample data were recorded for 6 inserts. The length of each sample was 4096 points, and these were acquired approximately in the middle of the bar. Each 4096 point record was processed to generate the features used in the classification stage. A total of 12 features were extracted from the sound and vibration data: absolute deviation, average, kurtosis, skewness and the energy in the frequency bands (2.2-2.4 and 4.4-4.6 kHz) obtained from the spectra. Two additional features were presented from the means of the feed and tangential forces. Results have shown that tool wear classification is difficult in the presence of such noisy data and it is therefore required that classification is made by a method that can resolve the complex interrelation between features to produce a robust wear classification. Also the use of multiple sensors should prove to be of great value towards tool wear evaluation since the noisy character of each sensor alone would lead to certain failure of the monitoring system, Silva et al. (2006). A typical graph of the evolution of flank wear with cutting time is shown in Figure 1 and consists of three stages. The first stage is a short period of rapid wear, the wear then progresses at a slower rate over a period, in which most of the useful tool life lies. The last stage is a rapid period of accelerated wear and it is usually recommended that the tool be replaced before this stage. The data presented in Figure 33 was obtained from a cutting speed of 350 m/min, a feed rate of 0.25 mm/rev, and a 1 mm depth of cut. For these cutting conditions the first stage can be observed to end at approximately 3.5 min after the start of cutting, which corresponds to VBB = 0.09 mm, the second stage lies in the interval between 3.5 min and 14.7 min (0.09 < VBB < 0.3 mm), and the third stage starts after 14.8 min of cutting time. The beginning of the third stage coincides with a value of flank wear of 0.3 mm which is the tool life criterion established in the ISO3685 (1993) standard.  Figure 1 – Flank wear evolution with time: 350 m/min, 0.25 mm/rev and 1 mm depth of cut As can be observed for the feed force, e.g. Figure 1, both tangential and feed forces show an increase with tool wear which is consistent between tools. These tests were carried out for 6 insert tips giving a total of 52 different wear levels. The 6 inserts gave a standard deviation of 2.1 min (14% of average tool life) justifying once again the use of a monitoring system. 5. Results and Discussion  Figure 2 – Average Mutual information of acquired Sensor data for new inserts From Figure 2 it can be seen that both forces, feed and tangential, present an estimated time delay of 0.1 ms. Sound achieves its first minimum at 0.2 ms and vibration shows its first minimum at 0.25 ms.  Figure 3 – Average Mutual information of acquired sensor data for worn inserts From Figure 3 it can be seen that for worn inserts Mutual Information changes significantly having both feed and tangential forces a delay time corresponding to 0.0001. Sound has its first minimum at 0.0001 seconds and vibration shows its first minimum 0.00015 seconds. The mutual information functions shown in Fig. 6(b) provide similar information to the autocorrelation coefficients about the behavior of both signals. The mutual information function of the differential pressure signal exhibits stronger gradient (sharper peak ~’= 0) than the mutual information function of the temperature signal. This is a strong indication that the differential pressure signal is more chaotic (or less predictable) than the temperature signal. This is to be expected since the ‘thermal inertia’ of the tube will provide a damped temperature signal, whereas the differential pressure measurement system will not damp differential pressure fluctuations at a frequency of 3 Hz [22]. After reaching its first minimum around ~-= 9, the differential pressure signal shows gain in information, again around a value of z=20. Note that the first minimum occurs at one half of the time for the first maximum, as expected.  Figure 4 – False nearest neighbours of acquired sensor data The FNN method provides a further evidence for the presence of low-order chaos in the time series for the present study. It is implemented by varying the values of the embedded dimension from 1 to higher values until the percentage of these false nearest neighbours drops to zero. The results in Fig. 5 show that the value of embedding dimension is 12 and 11 for stage and discharge data, respectively; and notably the values rapidly approach zero even when the value of the dimension reaches 5–7. The identification of these values means that both time series have an attractor, the geometric structure of which is unfolded as a distinct system whose orbits are distinct and do not cross (or overlap). The results obtained with the false nearest neighbour method are presented in Figure 4. It can be observed that all sensors signals fraction of false nearest neighbours convincingly drops to zero for m = 3. This means that the underlying dynamics of each sensed signals is a 3D system. In other words, it would be justified to model the behaviour of the system with not less than tree autonomous first-order ordinary differential equations. The largest Lyapunov exponent converges very well to max = 0.33. This is a firm proof for the chaotic behaviour of the studied experimental system.

Figure 5 – Recurrence plots for sound emissions of insert 1: (a) new tool; (b) worn tool flank wear of 0.3 mm; (c) worn tool flank wear of 0.36 mm

Figure 4 – Recurrence plots for sound emissions of insert 3: (a) new tool; (b) worn tool flank wear of 0.28 mm; (c) worn tool flank wear of 0.43 mm One possible candidate explanation might be tool chip friction. A friction model was used by Grabec (1986) in his pioneering paper on chaos in machining. It would rather be an academic exercise if chaos theory just proved the existence of a deterministic state without an ability to predict future patterns. To this end, it is encouraging that the evidence for its predictive ability is also confirmed by the results shown in Figs. 7–11. The following question may then be asked: does rating interfere with discharge values if the stage time series are influenced (or at explainable) by chaos theory)? As chaotic signals are detected in the Kizilirmak time series, the results presented in this paper show that the process of rating of stage to discharge time series amplifies inherent uncertainties and that these adverse impacts are attributable to inherent chaotic signals. This is a significant finding due to the importance of rating in open channel hydraulics. The significance of this finding stems from the fact that rating curves have wide applications and they all overlook this possible behaviour. Some of the implications are discussed below. If this finding is widespread, it may be necessary to devise correction schemes, details of which are not investigated in this paper. The predictive capability of the methods based on chaos theory into the future is formidable and in the case of the stage time series for the Kizilirmak, a reliable prediction penetrates deep into the future, as long as 2 years. This period for the prediction of discharge time series is reduced to 163 days (from 769 days), and this may be attributed to the interference of the underlying chaotic behaviour with the rating procedure. This predictive capability into the future is particularly important in various studies, where equivalent methods of design flood hydrographs have not been developed. For instance, water quality time series for river flows often have a record of years of observations on water quality parameters such as temperature, chlorophyll concentrations, dissolved oxygen and biochemical oxygen demand time series. Recorded values contain fluctuations creating difficulties to predict future values. There is no information if these fluctuations are stochastic or chaotic; however if they are chaotic, the predictive ability into the future makes an important research case for the application of the theory into forecasting water quality problems over time spans of 1 or 2 year durations. This paper investigates possible chaotic behaviours in the stage and discharge dynamics for the data recorded at Sogutluhan station, the Kizilirmak, Turkey. The analysis was performed on daily stage and discharge records over 8 years (1995–2002), where the values of discharge time series are obtained by rating the stage time series. The focus of the paper was on identifying chaotic signals in both stage and discharge time series with an immediate concern that if there were chaotic signals in the recorded stage values, how would they be carried (or propagated) into discharge values? This concern is of practical importance. The analysis was based on five widely used nonlinear dynamic methods: (1) Average Mutual Information to determine the delay time and reconstruct phase space; (2) False nearest neighbour algorithm and correlation dimension method to estimate the dimensionality; (3) Lyaponov exponent method and local approximation prediction for convergence/divergence, predictability, and prediction. The results from these methods provide convincing indication, cross-verification and confirmation of the existence of a mild low-dimensional chaos in both stage and discharge time series for the data used. For instance, clear and well-defined attractors in the phase space are observed; the correlation dimension values are less than 3; and the largest Lyapunov exponents are   6. Conclusion This paper described the implementation of a prototype decision support system for tool wear monitoring feature selection based on the self-organizing map. It was shown that the modelling technique proposed is highly effective for the classification of wear levels of tool inserts using apparently weak features. The results show that the self-organizing map neural network is a powerful tool for feature selection and validation as it performs vector quantization and hence feature contribution towards final classification can be analysed in a straightforward manner. Tests presented show a case study where this has been applied with success. The results obtained from the statistical and frequency parameters, as well as forces, are somewhat difficult to interpret considering them one at a time as some appear to correlate, whilst others appear to hold no correlation with tool wear. This can be overcome by taking into account the neural networks’ ability to extract information from apparently scattered information. The use of a Self Organising Map (SOM) structure has shown that classification was performed quite efficiently although the interpretation of results was not that easy, due to the complexity of the output structure. This work has illustrated the potential of Neural Networks when applied to tool wear monitoring. Further, it has enhanced the potential of neural networks, and in particular the self-organizing map, to perform tasks other than classification providing a insight view of feature value and potential towards data modelling. Whenever the dynamics of a system to be monitored and controlled is reconstructed from time-series signals, the solutions for the controller dynamics are obtained by integrating a system of differential equations (Hubler, 1991). The computational strategy developed from our understanding of the step-size-accuracy relationships will help in developing more accurate controllers. 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