Abstract
Abstract
Coordinate measuring machines (CMM) have a vital and enduring role in the manufacturing process because of their easy adaptation to the systems and high measurement accuracy. Owing to the demand for high accuracy and shorter cycle times of measurement tasks, determining the measurement errors has become more important in precision engineering. Additionally, manufactured components are becoming smaller and tolerances becoming tighter, and therefore, demands for accuracy are increasing. For this reason, dynamic error modeling has become a topic of considerable importance for improving measurement accuracy, manufacturing decisions and process parameter selections. A number of factors such as process parameters, measurement environment, measuring object, reference element, measurement equipment and set-up affect the measurement accuracy of CMM. Considering the complicated inter-relationships among a number of system factors, artificial intelligence-based techniques have become essential tools due to their speed, robustness and non-linear characteristics when working with high-dimensional data. In this study, a fuzzy logic-based methodology was implemented as an artificial intelligence approach for determining measurement errors related to the process parameters for coordinate measuring machines. A Mamdani-type fuzzy inference system was developed within the framework of a graphical user interface. Eight-level trapezoidal membership functions were employed for the fuzzy subsets of each model variable. The product and the centre of gravity methods were performed as the inference operator and defuzzification methods, respectively. The proposed prognostic model provided a well-suited method and produced promising results in predicting measurement errors by monitoring the process parameters such as optimum measuring point numbers, probing speed and probe radius.
Introduction
A coordinate measuring machine (CMM) is one of the most common inspection tools to determine the geometrical specification of the product. CMM technology is gradually increasing in light of their high accuracy and flexibility. With the vital and enduring role of CMM in the manufacturing process, the measurement accuracy becomes more important for determining the manufacturing product quality which has driven metrologists look for improved ways to perform the inspection of manufactured parts [1–3]. Expected product quality is more strongly affected by design decision and manufacturing procedures. Precisely determining product quality is also as crucial as the design and manufacturing steps. Today’s rapidly emerging industrial manufacturing quality requires strict dimensional and geometric controls to achieve high levels of functional performance. Form errors, geometrical imperfections and waviness errors have crucial influence on the functional performance of the manufactured product [4]. From this perspective, manufactured products must be accurately verified for conformance with design specifications. Every measurement procedure has a measurement uncertainty, which effects measurement accuracy [5]. Accurate inspection and defining measurement uncertainty correctly will enhance the product quality. The Guide to the Expression of Uncertainty in Measurement (GUM) provides internationally agreed upon approaches to the evaluation of measurement uncertainty [6].
In this context, many measurement strategies, analytical investigations, and new generation calibration equipments have been conducted in recent decades to evaluate measurement uncertainty to improve measurement accuracy. Analytical investigations highlight that the propagational distribution method of variables affects the measurement uncertainty. The GUM refers to three type of distribution methods such as Gaussian distribution, rectangular distribution, and U-distribution. According to the standard, the the U-distribution is identified as the most conservative assumption [7, 8]. The Monte Carlo method has been applied in some studies to analytical investigation to improvement of measurement accuracy [9, 10]. In an another study, different algorithms were used (summation of least-squares, linear least-squares and non-linear least-squares) to consider the data gathered from the spherical surfaces [11]. However, the variety of factors that affect measurement accuracy and uncertainy are not considered according to these distribution types, such as sampling methodology, measured part, and probe type. In some cases, sample size, product tolerance band, and measurement duration are also curicial for measurement uncertainty. Therefore, studies have also produced some new alternative methods for optimizing and predicting techniques using artificial intelligent analysis like virtual CMM methodology [12–14]. Virtual systems are applied to the CMM software, which enables the definition of measurement errors online during the actual measurement. Improving this software also provides a summary of the determinable component errors in the mathematical model.
Several authors claimed that the sampling number and method are among the substantial measurement uncertainty components, which are an important software-related issue in coordinate metrology [15–22]. The accuracy of measurement uncertainty decreases with the increase in the sampling number during the procedure. Surface waveness and topographical specifications are crucial parameters that affect uncertainty and accuracy according to the sampling interval. Studies used different sampling strategies to optimize the sampling number and method that involves a comprimise between the cost of inspection and the reliability of the measurement result [15–18]. An artificial neural network method-based was also proposed to optimize the sampling number and size according to the manufacturing method [19]. They obtained the implict correlation between the sample size and manufacturing technology of the sample by means of a backpropagation neural network with respect to the collected measurement data. As a result, it is declared that the neural network architecture determined the sampling size according to different manufacturing methodologies [19]. Furthermore, a fuzzy logic system was adapted to the software of the CMM measurement system. In one study, a fuzzy logic system was adapted to the software system to control the moving table of CMM. This system was a self-organizing fuzzy logic system that controls the dynamic table movement that affects measurement accuracy [23].
In addition to measurement methodology and software investestigations, there are some sophisticated additional equipment for the measurement procedure to improve measurement uncertainty and accuracy. A comprehensive review and assessment was made for the use of multisensor data fusion in dimensional metrology, which was employed to obtain holistic, more accurate, and reliable information [24]. Furthermore, an integrated laser interferometer system was used with the CMM system in order to improve the existing comparative procedure for calibrating internal dimensions [25]. In one recent study, a new generation touch probe was developed that consists of a reconfigurable stylus, an integrated force/torque transducer, a lower adapter, and an intelligent data acquisition system [26]. It can be noted that modeling CMM measurement error is very difficult because its performance is complex and varies significantly with conditions.
Fuzzy and neuro-fuzzy techniques have gained greater attention over the past decade. They have a distinctive advantage since these models provide a transparent and systematic analysis without requiring complex formulations and tedious parameter estimation procedures [27, 28]. Although much attention has been given to the importance of CMM measurement uncertainty considering the geometrical product specifications, to the best of the authors’ knowledge, there are no systematic papers in the literature specifically devoted to a study regarding an artificial intelligence-based modeling of the CMM measurement procedure for different probe diameter sizes, and selected measurement parameters used as inputs in the fuzzy logic technique.
Regarding the development of manufacturing technologies and the product demand for shorter cycle times of the measurement tasks, or demand for slower measurement with higher measurement accuracy, are increased requirements for the user. These selections are made in regard to procedure parameters. Shorter measurement cycle times results in faster probing velocity and lower probed point numbers. According to these parameters, the influence of the dynamic errors of the CMM system will increase. If the accuracy is the major demand for the user, dynamic error can be decreased with the proper selection of measurement parameters.
Consequently, based on the above-mentioned facts, it is noted that a knowledge-based prognostic modeling scheme may provide a transparent and systematic analysis for modeling the measurement procedure by a set of logical measurement parameter selections in a rapid and practical manner. Thus, in this paper, the development of an artificial intelligence-based modeling scheme by using the fuzzy logic methodology was proposed and described to provide the user a proper selection of procedure parameters according to demand. Considering the non-linear nature of CMM measurement errors, the specific objectives of this study were: (1) to develop a fuzzy-logic-based prognostic model that could be able to predict the measurement error according to the selected parameters; (2) to compare the proposed fuzzy-logic-based methodology to the conventional non-linear regression-based analysis for various descriptive statistical indicators, such as the coefficient of determination (R 2), mean absolute error (MAE), root mean square error (RMSE), index of agreement (IA), fractional variance (FV) and coefficient of variation (CV); and (3) to verify the validity of the proposed prognostic approach by several experimental data used as the testing set.
Experimental
Collection of the experimental data
In this study, the experiments were conducted using a conventional type of CMM (HERA SC 15.10.09), which was used to assess the conformance of the manufactured parts to the engineering drawing. This procedure was made in a non-conditioned and temperature controlled laboratory (temperature: 24(±2)°C). The CMM was calibrated and an interim check was made by an accreditated calibration service (CERMET, Italy). According to the report, the machine was ready for use with 3.7μm maximum permission error (upper limit: 4μm) due to ISO 10360-2:2009 [29]. After the calibration procedure, controlling the measurement error by means of optimizing measurement parameters can ensure using one kind of coordinate measuring machine for a big large variety of different geometries and sample sized manufacturing parts with lower measurement errors by means of controlling the effect of machine measurement parameters on measurement results.
In the present experimental study, a conventional type of coordinate measuring machine with three different sized probes were used. The sampling strategy was also defined with selected measurement velocities, approach distance, probed point number and proble angle. Additionally, all measurements were made under a controlled labratory conditions with constant temperature of 24(±2)°C. As seen in Table 1, the probe diameter, probed point number, approach distance to the specimen, measurement velocity and measurement angle were selected as input data for the present fuzzy logic model. The output data were determined as the difference between the CMM result and calibration certificate value of the spherical gauge diameter. These inputs were selected to include both dynamic errors and artifact errors (probe diameter). As highlighted in [29], CMM uncertainty standardization (ISO 10360-2:2009) does not fully assess their behavior during a measurement procedure with different speeds [30]. Furthermore, this standardization does not include inputs that were selected in our experimental research (Table 1).
Table 1 presents the procedure parameters that are used by CMM to measure the diameter of a certified spherical gauge. The reference spherical gauge diameter value was taken from its certificate. The difference between the CMM masurement result and the certificated diameter value was accepted as the CMM dynamic measurement error. These differences were used as output values for the experimental sets to develop a fuzzy logic model. Optimum experimental sets with a good combination of measurement parameters were crucial for our computational study. Therefore, a significant number of measurement procedures were applied to a calibrated spherical gauge with various parameters for modeling the measurement error by means of a fuzzy logic methodology and non-linear regression analysis-based studies (Table 1). The number of complete data points recorded for all seven variables was 623. Variations of the input components considered in the proposed prognostic approach are depicted in Fig. 1.
Fuzzy logic architecture
Fuzzy logic methodology is an artificial intelligence-based method that makes use of fuzzy sets and fuzzy ‘linguistic’ rules to incorporate this uncertainty into the model [31, 32]. By means of this methodology, mathematical calculations are relatively easier in linguistic terms instead of complicated equations used in the conventional methods when definining the complex qualitative relationships among the variables [33].
There are five steps of the fuzzy inference process. The first is the fuzzification of selected input variables. Determining these parameters has a crucial effect on the efficiency of the model. In the antecedent the fuzzy operator (AND or OR) is applied. Thereafter, implications from the antecedent to the consequent and aggregation of the consequents across the rules. Following these steps, the variables are defuzzified [34]. In this step, numerical inputs and outputs (crisp variables) are transformed into symbols (i.e. A, B, speed, point number, low, small, etc.) according to the corresponding degrees and numbers of specific membership functions used in the fuzzy inference system (FIS) [33, 34]. The input variables are always a crisp numerical value of the input variable (i.e. herein, the interval for measurement speed of probe is between 10 and 100 mm/sec) and the output is a fuzzified degree of membership in the qualifying linguistic set (always an interval between 0 and 1).
To apply the method, firstly the input variables are fuzzified, and the fuzzy operator (AND or OR) is applied to obtain a number that represents the result of the antecedent for a given rule in the second step. This number will then be applied to the output function. In the FIS, two built-in AND methods (min (minimum) and prod (product)) and two built-in OR methods (max (maximum) and probor (the probabilistic OR method)) are performed [34, 35].
In the third step, proper weighting (a number between 0 and 1) is applied to each rule, and the implication method is implemented. For the implication process, two built-in methods are basically supported by the FIS, and they are the same functions that are used by the AND method: min (minimum), which truncates the output fuzzy set, and prod (product), which scales the output fuzzy set [34, 35].
According to fuzzy logic theory, the decisions are being made according to the rules. In the fourth step, all of the rules are combined to make the decisions. For this purpose, aggregation is applied to fuzzy sets to conclude a single fuzzy set that represents the outputs of each rule. There are a number of aggregation methods (i.e. max (maximum), sum (simply the sum of each rule’s output set), probor, etc.) supported by the FIS [34, 35].
In the final step, the defuzzification procedure is applied to resolve a single output value from the set. In order to apply defuzzification technique, there are some methods such as center of gravity (COG or centroid), bisector of area, mean of maxima, leftmost maximum, and rightmost maximum, have been reported [34–36].
In this study, the product (prod) technique was selected for the inference operator since its performance was better in the collection of all the relationships among inputs’ and outputs’ fuzzy sets in the fuzzy rule base. Furthermore, the sum operator was used for the aggregation method conducted in the proposed FIS, as similarly performed in the previous studies of the second author [33, 38]. Additionally, the centre of gravity (COG or centroid) method was employed for the defuzzification technique as conducted in several fuzzy logic-based studies [33, 37–39]. According to the above-mentioned fuzzy logic application steps, a detailed schematic of the proposed knowledge-based prognostic modeling scheme to predict the dynamic measurement error of CMM is depicted in Fig. 2.
Selection of membership functions
The fuzzy membership function converts the variables between 0 and 1 to describe how each point in the input space is mapped to a membership value (or degree of membership) [40]. It was reported that triangular and trapezoidal shaped membership functions are predominant the in current applications of the fuzzy set theory, since their simplicity in both design and implementation is based on little information [41]. In this study, several combinations of triangular (trimf) and trapezoidal (trapmf) shaped membership functions were pre-trained at different levels (i.e. 3, 5, and 8) to investigate the best-fit fuzzy-logic model structure. The collected experimental data from the CMM program were randomly classified into different fuzzy set categories with respective minimum and maximum values of model variables.
Both triangular and trapezoidal membership functions were tested until the satisfactory outputs were achieved by the set of rules used in the fuzzy inference system with different scalar ranges of functions, as similarly conducted in the previous studies of the second author [33, 38]. For the present application, preliminary results of the computational analysis demonstrated that the optimum prediction performance in prediction of the measurement error of CMM was obtained with the use of trapezoidal shaped membership functions (trapmf) with eight levels for both input and output variables.
Fuzzification of input and output variables
In this study, the FIS (Fuzzy Inference System) Editor GUI (graphical user interface) in the Fuzzy Logic Toolbox within the framework of MATLAB ® 7.9.0.529 (Licence No: 161051, The MathWorks, Inc., USA, R2009b) software, running on a AMD Athlon(tm) II X3 460 CPU (Processor 3.40 GHz, 4 GB of RAM) PC, was employed for the modeling and simulation of dynamic error of CMM, according to selected input values. In the computational analysis, six input variables (probe diameter, probe velocity, measurement angle (x direction, °), measurement angle (y direction, °), probed point number, approach distance to probe) and the output variable (dynamic measurement error of CMM) were built using a Mamdani-type FIS Editor, and fuzzified with eight-level trapezoidal membership functions. Figure 3 shows the input and output variables in the MATLAB ® numeric computing environment.
Three different probes were used with different diameters: 1,2, and 4 mm. Figure 4(a) depicts the shape and range of each level for the first input variable (X 1). Probe velocity, the second input variable (X 2), ranged from 10 to 100 mm/sec, and the shape and range of its membership functions are shown in Fig. 4(b). The measurement angle (x direction) was considered as the third input variable (X 3), and ranged from 0° to 90°; measurement angle (y direction) was selected as the fourth input variable (X 4), and ranged from 0° to 75°, and the probed point number was taken as the fifth input variables (X 5), and ranged from 5 to 15 (Figs. 4(c), 5(a) and (b)). The approach distance, the sixth input variable (X 6), ranged from 4–15 mm. Figure 5(c) illustrates the shape and range of each level for the sixth input variable (X 6). The fuzzification of the dynamic measurement error being the output variable (Y) of the proposed fuzzy-logic model is shown in Fig. 6. Table 2 summarizes the number of trapezoidal membership functions (trapmf) and their fuzzification ranks, trapmf, for each of the input and output variables considered in the present fuzzy-logic-based model.
Fuzzy set categories were defined in the form of letters (i.e. A, B, C, etc.), to simplify the implemented rules, as similarly conducted in the previous works of Yetilmezsoy [33, 38]. In the present case, each model variable had eight trapezoidal shaped membership functions namely A, B, C, D, E, F, G and H, instead of long definitions such as moderately low, low, moderate, moderately high, high, very high, etc. For instance, according to the ranges and codes given in Table 2, an experimental set of “X 1= probe diameter = 4 mm X 2= probe velocity = 95 mm/s, X 3= measurement angle (x direction) = 45°, X 4= measurement angle (y direction) = 15°, X 5= probed point number = 6, X 6= approach distance to probe = 9 mm and Y= dynamic measurement error = 0.003μm” was coded as “A, C, B, D, E, F and B”, respectively. Furthermore, Table 3 presents the rule base of 20 example rule sets that were randomly selected from the overall fuzzy sets built within the framework of MATLAB ® software. On the basis of the present fuzzy set categories and the collected experimental data, a total of 99 rules were established in the IF-THEN format for the proposed fuzzy-logic model (trapezoidal shaped membership functions with eight levels for each of the input and output variables) structures by using the Fuzzy Rule Editor.
Non-linear regression analysis-based modeling
In this study, a multiple regression-based model was also derived to appraise the dynamic measurement error of CMM in addition to the fuzzy-logic approach. For the comparative purpose, the experimental data were evaluated by a licensed multiple regression software package (DataFit ® V9.0.59, Oakdale Engineering, PA), containing 298 two-dimensional (2D) and 242 three-dimensional (3D) non-linear regression models. The regression analysis was performed based on the Levenberg-Marquardt method with double precision, as was similarly done in several studies of the second author [33, 42–46].
The CMM-based data were imported directly from Microsoft ® Excel, which was used as an open database connectivity data source, and then the non-linear regression analysis was implemented. As regression models were solved, they were automatically sorted according to the goodness-of-fit criteria into a graphical interface on the DataFit ® numeric computing environment. In the analysis, the regression variables (β 1, β 2, β 3, β 4, β 5, and β 0), standard error of the estimate (SEE), coefficient of multiple determination (R 2), adjusted coefficient of multiple determination (), the number of non-linear iterations (NNI) were computed to evaluate the performance of the regression models. For the appraisal of the significance of the regression coefficients, t-ratios and the corresponding p-values were also calculated. To determine the statistical significance of the model, an alpha (α) level of 0.05 (or 95% confidence) was used.
Measuring the goodness of the fit
In this study, various important statistical indicators such as coefficient of determination (R 2), mean absolute error (MAE), root mean square error (RMSE), index of agreement (IA), fractional variance (FV) and coefficient of variation (CV) were computed as helpful mathematical tools to quantify the fit between the experimental data and the model outputs. Some of these estimators were also tested by using a licensed statistical software package (StatsDirect (V2.8.0, Copyright © 1990–2011, Stats Direct Ltd.) for the statistical validation of the calculated values. Detailed definitions of these descriptive statistics can be found in several studies [47–49].
Results and discussion
Prediction of dynamic measurement error of CMM
In the present study, an artificial intelligence approach based on the fuzzy logic methodology and non-linear regression analysis were conducted to predict the dynamic measurement error of CMM according to the selected input variables. In the non-linear regression analysis, one exponential model and one first-order polynomial model were also derived for the estimation of the dynamic measurement error. The results of the non-linear regression analysis are summarized in Table 4. Regression variable results including the standard error of the corresponding p-values, the t-statistics and the estimate (SEE) for the best-fit regression model (herein the exponential term) are given in Table 5. The exponential model, which was derived as a function of five input variables [Y= Dynamic measurement error = f(probe diameter (X
1= d), probe velocity (X
2= V), measurement angle of probe (X
3,4= x°, y°), probed point number (X
5= N), approach distance to the gauge (X
6= D)] is expressed as follows:
According to the literature, the larger t-ratio indicates the more significant parameter in the regression model. Additionally, the variable with the lowest p-value is considered the most significant [33, 44]. As seen in Table 5, the resulting t-ratios, the probed point number, and the approach distance to the probe have more importance than other variables for the derived exponential model in the prediction of the dynamic measurement error. It is noted that mechanical characteristics of the present model variables are fully discussed in previous studies [1–5].
Figure 7 shows a head-to-head comparison of performances for the multiple regression-based models on the prediction of the dynamic measurement error. Although the exponential model (non-linear regression model (NRM–1) produced smaller deviations compared to the first-order polynomial model (NRM–2 with constant term), the non-linear regression-based methodology showed poor prediction performance on the experimental data with high residual errors. Considering the overall performances, the conventional regression analysis approach did not produce satisfactory predictions of the dynamic measurement error as good as the proposed fuzzy logic-based model (Fig. 8).
In this experimental study, different statistical model outputs were assessed with various descriptive statistics such as R 2, MAE, RMSE, IA, FV and CV for measuring the predictive accuracy. To validate the predictive capability of the prediction models, additional measurements applied with the CMM machine were used as the testing set for both the fuzzy logic model and non-linear regression model. Numerical results are summarized in Table 6. According to these testing outputs and deviations of the developed models, it can be concluded that the proposed fuzzy logic model demonstrated a very satisfactory performance in the prediction of the dynamic error of CMM compared to the non-linear regression-based approach (Fig. 9).
As seen in Table 6, descriptive performance indices revealed that the fuzzy logic-based model produced very small residual errors and demonstrated a superior predictive performance compared to the best-fit non-linear regression model. In the prediction of the dynamic measurement error of CMM, the values of the determination coefficient (R 2= 0.971 for overall data and R 2= 0.978 for testing data) indicated that only about 2.9% and 2.2% of the total variations were not explained by the fuzzy logic model for the overall data set and the testing data set, respectively. However, for the exponential model, about 17.2% and 24.4% of total variations did not fit the experimental data for the overall data set (R 2= 0.828) and the testing data set (R 2= 0.756), respectively.
Based on the testing data set, the values of fractional variance ((FV)FLM = 0.1034 and (FV)NRM = 0.1207) revealed that the fuzzy logic model demonstrated greater accuracy than the non-linear regression approach in the prediction of the dynamic measurement error of CMM, respectively. Moreover, the values of the mean absolute error ((MAE)FLM = 0.000616 and (MAE)MRM = 0.00175) and the values of the root mean squared error ((RMSE)FLM = 0.000728 and (RMSE)MRM = 0.00209) also indicated that the proposed fuzzy-logic model performed better than the conventional non-linear regression-based method. Furthermore, values of index of agreement ((IA)FLM = 0.992 and (IA)MRM = 0.926) concluded that the proposed an artificial intelligence-based model was the most accurate prediction model to predict the dynamic measurement error of CMM for the present case. Low values of the coefficient of variation (CV = 9.901% for overall data and 10.304% for testing data) obtained by the fuzzy logic model also indicates a very high degree of precision and a good deal of the reliability of the experimental data, as similarly reported by Yetilmezsoy et al. [49]. On the basis of the above-mentioned statistical results, this experimental study has obviously indicated the potential of the proposed artificial intelligence-based approach for capturing the complicated inter-relationships between the dynamic measurement error and other design factors.
Conclusions
A fuzzy logic-based artificial intelligence approach was conducted as an important objective to develop a prognostic modeling scheme that could make a reliable prediction about the dynamic measurement error for CMM based on the selected parameters and probe (inputs). The following conclusions can be withdrawn from the present study:
Statistical indicators revealed that the proposed prognostic approach based on the fuzzy-logic methodology produced much smaller deviations and demonstrated a superior predictive performance compared to the conventional non-linear regression-based method. According to the CMM working parameters, the proposed model will aid users in the selection of lower measuring points or higher measuring velocities based on the required measurement accuracy in a cost-effective manner, providing shorter measurement times. The artificial intelligence methodology can provide a well-suited alternative to optimize machine working parameters according to the sample and accuracy needs. Moreover, this method can be used during the calibration procedure to understand the effects of the relevant parameters on the measurement accuracy for shorter calibration times. The proposed knowledge-based prognostic modeling scheme is very easy to use and there is no need to define complex and time-consuming mathematical formulations to predict the dynamic measurement error. Since the fuzzy logic model has a high capability of capturing the dynamic behavior and complicated inter-relationships between multi-input and output variables, the measurement procedure error behavior could be easily modeled in a highly non-linear system. Therefore, it is believed that the fuzzy-logic technique can be practically used by adapting it to different factor levels for the modeling of other measurement procedures. The results clearly showed that the fuzzy logic-based model could be utilized as an easy-to-use mathematical tool for the prediction of digitization uncertainty of the measurement procedures during standard measurement or calibration procedures.
Footnotes
Acknowledgments
The authors would like to thank to the Applied Automation Technologies, Inc. (Istanbul, Turkey) for their CMM support. The authors declare that there are no conflicts of interest including any financial, personal, or other relationships with other people or organizations.