1619
Journal of Intelligent & Fuzzy Systems 29 (2015) 1619–1633
DOI:10.3233/IFS-151641
IOS Press
Application of fuzzy logic methodology
for predicting dynamic measurement errors
related to process parameters of coordinate
measuring machines
Asli G. Bulutsuza,∗ , Kaan Yetilmezsoyb and Numan Durakbasac
a Department
of Mechanical Engineering, Faculty of Mechanical Engineering, Yildiz Technical University,
Besiktas Campus, Besiktas, Istanbul, Turkey
b Department of Environmental Engineering, Faculty of Civil Engineering, Yildiz Technical University,
Davutpasa Campus, Esenler, Istanbul, Turkey
c Department for Interchangeable Manufacturing and Industrial Metrology, Institute for Production Engineering
and Laser Technology, Vienna University of Technology, Austria
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.
Keywords: Coordinate measuring machines, fuzzy logic, measurement accuracy, uncertainty
1. Introduction
∗ Corresponding
author. Asli G. Bulutsuz, Department of
Mechanical Engineering, Faculty of Mechanical Engineering, Yildiz
Technical University, 34349, Besiktas Campus, Besiktas, Istanbul,
Turkey. Tel.: +90 212 383 29 59; Fax: +90 212 383 30 24; E-mails:
gunay@yildiz.edu.tr; asligunaya@gmail.com.
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
1064-1246/15/$35.00 © 2015 – IOS Press and the authors. All rights reserved
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A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
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
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A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
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 intelligencebased 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 (R2 ), 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.
2. Experimental
2.1. 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 3.7 m maximum permission error
(upper limit: 4 m) 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.
Table 1
Summary of the parameters used in the present study
Probe diameter (X1 )
1,2,4 mm
Probe velocity (X2 )
Measurement angle (X3 and X4 )
Probed point number (X5 )
Approach distance (X6 )
10–100 mm/s
A0◦ A90◦ - B0◦ B75◦
5–15
4–15 mm
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Fig. 1. Variation of the input components: Probe diameter (mm) (X1 ), probe velocity (mm/s) (X2 ), measurement angle (x direction, ◦ ) (X3 ),
measurement angle (y direction, ◦ ) (X4 ), probed point number (X5 ), and approach distance (X6 ).
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 nonlinear 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.
2.2. Fuzzy logic architecture
Fuzzy logic methodology is an artificial intelligencebased method that makes use of fuzzy sets and fuzzy
‘linguistic’ rules to incorporate this uncertainty into
A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
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
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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, 37, 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, 34, 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.
2.3. 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, 34, 37, 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.
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A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
Fig. 2. A detailed flowchart of the CMM fuzzy-logic methodology conducted in this study.
2.4. 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 (X1 ).
Probe velocity, the second input variable (X2 ), 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 (X3 ), and ranged from 0◦ to 90◦ ;
measurement angle (y direction) was selected as the
fourth input variable (X4 ), and ranged from 0◦ to 75◦ ,
and the probed point number was taken as the fifth input
A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
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Fig. 3. Input and output variables considered for the proposed fuzzy inference system (FIS).
variables (X5 ), and ranged from 5 to 15 (Figs. 4(c), 5(a)
and (b)). The approach distance, the sixth input variable
(X6 ), ranged from 4–15 mm. Figure 5(c) illustrates the
shape and range of each level for the sixth input variable
(X6 ). 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[a b c d],
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, 37, 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 “X1 = probe diameter = 4 mm X2 =
probe velocity = 95 mm/s, X3 = measurement angle (x
direction) = 45◦ , X4 = measurement angle (y direction) = 15◦ , X5 = probed point number = 6, X6 =
approach distance to probe = 9 mm and Y = dynamic
measurement error = 0.003 m” 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.
2.5. 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, 34, 42–46].
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A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
Fig. 4. Fuzzification of CMM-based parameters (a) probe diameter
(X1 ), (b) probe velocity (X2 ), and (c) measurement angle (x direction)
(X3 ).
Fig. 5. Fuzzification of CMM-based parameters (a) measurement
angle (y direction) (X4 ), (b) probed point number (X5 ), and (c)
approach distance to probe (X6 ).
Fig. 6. Fuzzification of dynamic measurement error (Y ).
[−0.0024 −0.002 0.002 0.0024]
[0.002 0.0024 0.0041 0.0049]
[0.0041 0.0049 0.0063 0.0072]
[0.0063 0.0072 0.0087 0.0098]
[0.0087 0.0098 0.0106 0.0121]
[0.0106 0.0121 0.013 0.0142]
[0.013 0.0142 0.0159 0.017]
[0.0159 0.017 0.019 0.0201]
[1 2 4 5]
[4 5 5.5 6.5]
[5.5 6.5 7 8]
[7 8 8.5 9.5]
[8.5 9.5 10 11]
[10 11 11.5 12.5]
[11.5 12.5 13 14]
[13 14 16 17]
[3.4 4 6 6.6]
[6 6.6 7 8]
[7 8 8.5 9]
[8.5 9 10 11]
[10 11 11.2 12]
[11.2 12 12.4 13]
[12.4 13 14 14.5]
[14 14.5 15.5 16]
[−12 −5 5 12]
[5 12 14 22]
[14 22 26 32]
[26 32 36 42]
[36 42 49 52]
[49 52 55 62]
[55 62 68 73]
[68 73 77 82]
[−15 −10 10 15]
[10 15 20 26]
[20 26 30 38]
[30 38 42 50]
[42 50 55 65]
[55 65 70 76]
[70 76 80 86]
[80 86 94 100]
Output variable
Y
Dynamic measurements
error
(m)
X6
Approach distance
to probe
(mm)
X5
Probed
point
number
Input variables
X3
X4
Measurement angle
Measurement angle
(x direction)
(y direction)
(degree)
(degree)
1627
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 (β1 , β2 , β3 , β4 , β5 , and
β0 ), standard error of the estimate (SEE), coefficient
of multiple determination (R2 ), adjusted coefficient of
multiple determination (R2a ), 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 (␣) level of 0.05 (or 95% confidence) was used.
2.6. Measuring the goodness of the fit
In this study, various important statistical indicators
such as coefficient of determination (R2 ), 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].
[0.5 0.8 1 1.5]
[1.2 1.5 1.7 1.9]
[1.7 1.9 2 2.4]
[2 2.4 2.5 2.8]
[2.5 2.8 2.9 3.2]
[2.9 3.2 3.3 3.5]
[3.3 3.5 3.7 3.9]
[3.7 3.9 4.1 4.3]
A
B
C
D
E
F
G
H
[−4 5 15 24]
[15 24 30 36]
[30 36 40 48]
[40 48 55 62]
[55 62 69 72]
[69 72 79 85]
[79 85 90 95]
[90 95 105 110]
X1
Probe
diameter
(mm)
X2
Probe
velocity
(mm/s)
3. Results and discussion
Level of
trapezoidal
membership
functions
(trapmf)
Table 2
Number of trapezoidal membership functions (trapmf) and their ranks for each of the input and output variables considered in the present fuzzy sets
A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
3.1. 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
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A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
Table 3
A random selection of 20 example rule sets from the total 99 sets
Number of
fuzzy
rule
1
7
10
13
19
22
26
29
33
35
38
43
50
64
70
75
77
82
90
99
X1
Probe
diameter
(mm)
X2
Probe
velocity
(mm/s)
X3
Measurement angle
(x direction)
(degree)
H
H
H
H
H
H
H
H
H
A
A
A
A
C
C
C
C
C
C
C
A
A
A
D
D
H
D
H
H
A
A
A
H
A
A
D
D
H
D
A
A
A
H
A
B
D
A
D
H
D
H
A
A
H
A
A
H
D
D
D
Input variables
X4
Measurement angle
(y direction)
(degree)
X5
Probed point
number
X6
Approach distance
to probe
(mm)
Output variable
Y
Dynamic measurement
error
(m)
A
D
D
D
A
A
H
A
H
A
A
H
D
H
D
H
H
D
A
A
B
E
B
B
H
D
E
E
E
B
E
H
B
E
B
E
H
B
C
A
A
C
C
B
A
B
D
A
D
B
B
E
C
H
D
F
G
D
A
A
A
A
A
A
D
B
A
E
A
E
A
A
A
A
A
A
A
E
G
B
Table 4
Summary of the multiple regression-based results
Rank
Regression model
SEE
Dynamic measurement error (Y)
exp(aX1 + bX2 + cX3 +
1
dX4 + eX5 + . . . + fX6 + g)
2
aX1 + bX2 + cX3 + dX4 + eX5 + fX6
SR
2.60 × 10–18
0.0017
0.0018
RA
–0.0104
2.92 × 10–20
–0.00012
RSS
R2
R2a
NNI
0.00025
0.828
0.815
4
0.811
0.799
11
0.00027
SEE, standard error of the estimate; SR, sum of residuals; RA, residual average; RSS, residual sum of squares;
determination; R2a , adjusted coefficient of multiple determination; NNI, number of non-linear iterations.
R2 ,
coefficient of multiple
Table 5
Model components and regression variable results for the best-fit (exponential) model
Independent and original variables
SEa
t-ratio
p-valueb
1.492 × 10–4
5.469 × 10–6
5.176 × 10–6
8.609 × 10–6
5.085 × 10–5
4.898 × 10–5
8.286 × 10–4
–4.734
–1.172
1.736
–1.671
16.801
5.487
–2.834
0,00001
0.24429
0.08614
0.09845
0.00000
0.00000
0.00577
Y = exp (aX1 + bX2 + cX3 + dX4 + eX5 + fX6 + g)
Y = exp[(−7.06 × 10−4 )X1 − (6.41 × 10−6 )X2
+(8.99 × 10−6 )X3 − (1.44 × 10−5 )X4
+(8.54 × 10−4 )X5 + (2.69 × 10−4 )X6 − 0.00235]
X1 = Probe diameter (mm)
X2 = Probe velocity (mm/s)
X3 = Measurement angle (x direction) (degree)
X4 = Measurement angle (y direction) (degree)
X5 = Probed point number
X6 = Approach distance to probe (mm)
g = Constant term
a Standard
error. b p-values <0.05 were considered to be significant.
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 (X1 = d), probe
velocity (X2 = V), measurement angle of probe
(X3,4 = x◦ , y◦ ), probed point number (X5 = N), approach
distance to the gauge (X6 = D)] is expressed as follows:
A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
Y = exp(aX1 + bX2 + cX3 + dX4 +
eX5 + . . . + fX6 + g)
(1)
Y = exp[(−7.06 × 10−4 )X1 − (6.41×10−6 )X2 + . . .
(8.99 × 10−6 )X3 − (1.44 × 10−5 )X4 + . . .
(8.54 × 10−4 )X5 + (2.69 × 10−4 )X6 − 0.00235]
(2)
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, 43, 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].
Fig. 7. A head-to-head comparison of performances for measured
data, fuzzy-logic outputs and the multiple regression models (exponential model and the first order polynomial model with constant
term) by means of the dynamic measurement error (n = 89).
1629
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).
3.2. Appraisal of predictive accuracy
In this experimental study, different statistical model
outputs were assessed with various descriptive statistics such as R2 , MAE, RMSE, IA, FV and CV for
measuring the predictive accuracy. To validate the
Fig. 8. A head-to-head comparison of performances for measured
data, fuzzy-logic outputs and the best-fit regression model (exponential model) outputs by means of the dynamic measurement error
(n = 89).
1630
A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
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 (R2 = 0.971 for overall
data and R2 = 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 (R2 = 0.828) and the testing data set
(R2 = 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
Fig. 9. A head-to-head comparison of performances for measured
data, fuzzy-logic testing outputs (responses for 30 additional observed
data) and the exponential model outputs by means of the dynamic
measurement error (n = 30).
Table 6
Descriptive statistical performance indices for the data sets considered in the present prognostic approach
Performance indicators
Determination coefficient (R2 )
Calculationa
R2 =
n
n
(Oi −Om )2
Root mean squared error (RMSE)
Index of agreement (IA)
MAE =
1
n
RMSE =
IA = 1 −
Additional testing data (n = 30)
NRMc
FLMb
NRMc
0.971
0.828
0.978
0.756
0.00598
0.00132
0.000616
0.00175
0.000698
0.00167
0.000728
0.00209
0.998
0.992
0.992
0.926
0.0871
0.1034
0.1207
(Oi −Om )(Pi −Pm )
i=1
i=1
Mean absolute error (MAE)
2
Overall data (n = 89)
FLMb
n
i=1
1
n
n
(Pi −Pm )2
i=1
|Pi − Oi |
n
[Pi − Oi ]2
i=1
n
0.5
(Pi −Oi )2
i=1
n
(|Pi −Om |+|Oi −Om |)2
i=1
Fractional variance (FV)
a O,
FV = 2 σo − σp / σo + σp
0.0702
P and m are the subscripts indicating the observed, predicted and mean, respectively.
b Fuzzy-logic
model.
c Non-linear
regression model.
A.G. Bulutsuz et al. / Application of fuzzy logic methodology for predicting dynamic measurement
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.
1631
r The proposed knowledge-based prognostic mod-
r
eling 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 logicbased 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.
Acknowledgments
4. 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:
r Statistical indicators revealed that the proposed
r
r
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 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.
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