Efficient evaluation of flatness error from Coordinate Measurement Data using Cuckoo Search optimization algorithm

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Efﬁcient evaluation of ﬂatness error from Coordinate Measurement Data
using Cuckoo Search optomisation algorithm
Article · June 2019
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‫األكاديمية‬ ‫البحوث‬ ‫مجلة‬Journal of Academic Research June(2019) 37-51
37
Efficient evaluation of flatness error from
Coordinate Measurement Data using Cuckoo
Search optomisation algorithm
Ali M Abdulshahed1
Ibrahim Badi2
Ahmed Alturas1
1- Electrical and Electronic Engineering Department, Misurata University,
Misurata, Libya
2- Mechanical Engineering Department, Misurata University, Misurata, Libya
Abstract
In this work, an attempt has been taken to display a brief idea about the applications
of Nature-inspired optimisation algorithms in automated manufacturing systems. A
new technigue based on the use of the Cuckoo Search optomization algorithm for
flatness error estimation is proposed. The proposed technique has been validated and
compared with will known optimisation methods, including deterministic and
stochastic algorithms. Extensive simulation using Matlab environment in conjuction
with measured data has been carried out to show and choose the most suitable and
efficient algorithms for a given optimisation task. The analysis results for Cuckoo
Search are compared with those obtained by Particle Swarm Optimisation , Convex
hull, Improved Convex hull, and Least squares. The implementation proves that the
nature-inspired optimisation algorithms outperform traditional algorithms with good
global convergence capability and can act as an alternative optimisation method for
automated manufacturing problems.
Keywords: CMM, AI, nature-inspired optimisation, Cuckoo Search.
Paper type: Research paper
Introduction
Many efforts have been done in order to perform how a workpiece will be
manufactured with a good quality. Optimisation in manufacturing is an
important issue because it may result in a shorter machining time, better
surface quality and increase productivity. Recent growing interest in quality of
manufacturing process has heightened the need for suitable optimisation
techniques. Nature-inspired optimisation algorithms can be used for the
optimisation of the machining process, also for prediction of the parameters
of machining. For instance, researchers have adapted several (AI) methods for
tool path optimisation such as, Artificial Neural Network (ANN), and Genetic
Algorithms (GA). Although, these algorithms produce good solutions [1-3],
they do not ensure that an optimal path will ever be found at the price of a
prohibitive cost in computation. Brief details are given below of some

Efficient evaluation of flatness error from Coordinate Measurement Data using Cuckoo
38
common optimisation algorithms used in automated manufacturing systems.
References are given either to the work that proposed them or to a more
recent discussion of their use.
The classification of optimisation methods is not well established in literature,
especially about the use of some terminologies. Generally, classification can
be carried out in terms of the number of objectives, number of constraints,
function forms, landscape of the objective functions, type of design variables,
uncertainty in values, and computational effort. Classification of optimisation
algorithms can also be carried out in a simple way; deterministic algorithms
and stochastic algorithms.
Deterministic algorithms take the advantages of the analytical properties of
the problem to generate a sequence of steps that finally converge to an
optimum solution which might not be the global one. Deterministic algorithms
will follow the same procedure or path whether the algorithm runs today or
tomorrow. In the literature, several deterministic techniques exist such as
Convex hull, and Least squares. On another hand, stochastic algorithms always
have randomness in their procedure to find the optimal solution and classified
as the most recent and powerful computational products of artificial
intelligence techniques. Particle Swarm Optimisation (PSO) is a good example;
the solution will be different each time, though the final solution may be no
big difference, but the path of each particle is not exactly the same. Most of
classical algorithms are deterministic. For instance, simplex method in linear
programing is deterministic. Other deterministic algorithms used gradient
(gradient based algorithms) information such as Newton-Raphson algorithm,
steepest descent method, and conjugate gradient method. However, if there
is some discontinuity in the objective function, they do not work well [4].
There are two types of stochastic algorithms; heuristic and meta-heuristic.
Heuristic means “to find” or “to discover by trial and error”. Meta means
“beyond” or “higher level”. Generally, meta-heuristic algorithms are usually
considered as a higher level of heuristics, because meta-heuristic algorithms
are not simple trial and error, meta-heuristics are designed to learn from past
solutions, to be biased toward better moves, to select the best solutions, and
to construct sophisticated search moves. Therefore, meta-heuristics can be
much better than heuristic [4]. The efficiency of an algorithm can depend on
many factors, such as the intrinsic structure of the algorithm, the way it
generates new solutions, and the setting of its algorithm-dependent
parameters.

39
Recentelly, Genetic algorithm (GA) and Particle Swarm Optimisation (PSO) are
two popular intelligent computation techniques that have attracted much
attention in tool path optimisation work [2, 3]. However, GA often suffers
from premature convergence and degradation efficiency because of its highly
epistatic objective functions, which makes the identified parameters highly
correlated [5]. Even the crossover and mutation operations cannot ensure
better fitness of offspring, due to the similar structures of chromosomes in the
population.
The PSO algorithm was introduced by Kennedy and Eberhart in [6] as an
alternative to other evolutionary techniques such as GA. The PSO algorithm is
inspired by the behaviours of natural swarms, such as the formation of flocks
of birds and schools of fish. The advantages of the PSO algorithm is that it
does not require the objective function to be differentiable as in the gradient
decent method, which makes few assumptions about the problem to be
solved. Furthermore, it has a simple structure and its optimisation method
illustrates a clear physical meaning. PSO consists of a population formed by
individuals called particles, where each one represents a possible solution of
the problem. Each particle i has a position Xi and tries to search the best
position with time in D-dimensional space (solution space). PSO assumes all
particles to fly with velocity Vi that is continuously adjused in light of its own
experience and its companions’ experience, including the current position,
velocity and the best previous position experienced by itself and its
companions. The motion of each particle can be determined by the following
equations:
)()( 2211
1 k
in
k
jn
k
in
k
in
k
in
k
in XGrcXPrcVV 

(1)
11 
 k
in
k
in
k
in VXX
(2)
Where k is the iteration number, ω is the inertia weight, n=(1, 2, …. N), r1 and
r2 are rondom numbers between 0 and 1 standing for the weight that particle
gives to its own best position and that for its best neighbour's position,
Pin=(pi2, pi2, …, piN) is the best previous position of the ith
particle (that gives
the best fitness value), and Gjn=(gj2, gj2, …, gjN) is the global best position of the
best particle (J) in the swarm, c1 and c2 are accelerating coefficient that
determine the maximum position step size of the particle in a single iteration
[6, 7].
Therefore, instead of using the original algorithms, several versions of the PSO
algorithm have been proposed in the literature in order to to optimise the
automated manufacturing tasks and improve the performance of the original

40
algorithm [8]. Many researches have employed PSO for generation of
optimised tool path. Chih-Hsing et al., [2] Used PSO optimisation technique for
finding the optimal tool path in 5-axis flank milling of ruled surfaces. A later
work [9] used an improved tool path planning method based on PSO
algorithms by offering smaller machining error and better planning flexibility.
However, the right selection of PSO parameters plays an important role in
balancing the global search and local search [10]. In the abovementioned
studies, a better solution might be missed when these values are set fixed.
Nowdays, an adaptive strategy for tuning PSO parameters also has been used
in order to improve the performance of the tool path planning [11].
Cuckoo Search (CS) has been applied in many fields of optimisation and
computational intelligence with promising efficiency. Yildiz in [12] has used CS
to select optimal machine parameters in milling operation. The results were
compared with those obtained using other well-known optimisation
techniques such as, ant colony algorithm and genetic algorithms [12, 13]. The
obtained results demonstrated that the CS is an effective approach for the
optimisation of machining optimisation problems. More recently, Huang et al
[14] used a new hybrid algorithm named teaching-learning-based cuckoo
search (TLCS) for parameter optimisation problems in structure designing as
well as machining processes. Optimisation of drill path can lead to a significant
reduction in machining time which improves productivity of manufacturing
systems. Lim et al. [15, 16] reported a combinatorial cuckoo search algorithm
for solving drill path optimisation problem. The performance of CS algorithm
was tested and verified with three different case studies from the literature.
The simulation results conducted in this research indicates that the CS
algorithm was capable of finding the optimal path for holes drilling process.
From the algorithm analysis point of view, a conceptual comparison of CS with
Differential Evolution (DE), PSO, and artificial bee colony (ABC) in [17]
suggested that CS and DE algorithms provide more robust results than PSO
and ABC. Gandomi et al. [13] provided a more extensive comparison study for
solving various sets of structural optimisation problems and concluded that CS
in combination with Levy flights obtained better results than other algorithms
such as PSO and GA. Generally, the choice of an algorithm for an optimisation
task will largely depend on the type of the problem, the nature of the
algorithm, the desired quality of solutions, the available computing resources,
time limit, availability of the algorithm implementation, and the expertise of
the decision makers.
In this paper, an attempt has been taken to display a brief idea about the
optimisation algorithms, mostly the nature-inspired optimisation algorithms.

41
Extensive simulation using Matlab tests have been carried out to show and
choose the most suitable and efficient algorithms for a given optimisation
task. This work will enable the reader to open the mined to explore possible
applications in the field of automated manufacturing systems.
Methodology
Cuckoo Search (CS)
Cuckoo Search (CS) is a meta-heuristic algorithm, introduced in 2009 by Xin-
She Yang [18]. It has many advantages due to its simplicity and efficiency in
solving highly non-linear optimisation problems with practical engineering
applications [19]. CS satisfies the global convergence requirements and
supports local and global search capabilities. In addition, CS uses Lévy flights
based on the breeding strategy of some cuckoo species as a global search
strategy [20]. CS is a stochastic algorithm, inspired by natural behaviour of a
family of birds called Cuckoos. Some species of the cuckoo birds engage in an
aggressive reproduction strategy; they lay their eggs in the nests of other host
birds, which act as surrogate parents. The host bird may notice that the eggs
are not their own so it either throws them away or abandons the nest and
builds a new one elsewhere. Consequently, Cuckoo eggs have to be incredibly
good mimics in order to be accepted into the nest.
In brief, the CS algorithm for global optimisation is based on three rules [4]: (i)
each artificial cuckoo lays an egg in a randomly chosen nest in one generation;
(ii) nests, which have the high-quality eggs (solutions) will be retained to the
next generation; and (iii) the total number of nests is fixed, and a host species
can discover an exotic egg with a probability pa ∈ [0, 1]. Thus, the host bird
can either throw the egg away or abandon the nest, and then randomly build
a completely new nest in somewhere else. For simplicity in describing the CS
algorithm, this last assumption can be estimated by the fraction of pa of the n
nests that are replaced by new nests with new random solutions at new
locations. The fitness function of the solution is defined in a similar way as in
meta-heuristics evolutionary methods. It is worth pointing out that in this
simple algorithm, there is no distinction between a cuckoo, an egg, or a nest,
since each nest has a single egg. The aim is to use the new and potentially
better solutions to replace worse solutions that are in the nests. Based on
these three rules, the basic steps of the CS are described in a pseudo code
below:

42
Algorithm 1. Pseudo code of Cuckoo Search
(CS) [18]
1:Objective function: 𝒇( 𝑩), 𝑩 =
(𝒃𝒊𝟏, 𝒃𝒊𝟐, … , 𝒃𝒊𝑫) 𝑻
;
2: Generate an initial population of 𝒏 host
nests 𝒃; 𝒊 = 𝟏, 𝟐, … , 𝑴;
3: While ( 𝒕 < 𝑴𝒂𝒙𝑮𝒆𝒏𝒆𝒓𝒂𝒕𝒊𝒐𝒏) or (stop
criterion)
4: Get a Cuckoo randomly (say, 𝒊)
5: Generate a new solution by performing
Levy flights;
6: Evaluate its fitness 𝒇𝒊
7: Choose a nest among 𝒏 (say, 𝒋)
randomly;
8: If (𝒇𝒊 > 𝒇𝒋)
9: Replace 𝒋 by new solution
10: end if
11: A fraction ( 𝒑 𝒂) of worse nests are
abandoned and new ones are built;
12: Keep the best solutions/nests;
13: Rank the solutions/nests and find the
current best;
14: Pass the current best solutions to the
next generation;
15: end while;
16: post process results;
17: end
Case study
The Coordinate Measuring Machines (CMMs) have proven to be reliable,
flexible and very much suitable for determining the acceptability of
manufactured parts. In recent years, CMMs have gained popularity in

43
automated inspection for both the on-line and off-line inspe ction of
manufactured components. The data for the evaluation of form errors
obtained from CMM will be in Cartesian coordinates given with reference to a
system of mutually orthogonal planes and the data combines form and size
aspects. This data has to be further processed using appropriate techniques to
evaluate the form error. CMM measurement uncertainty because of software
has been problematic in the past, and has the ability to be a continued source
of uncertainty, especially for minimum circumscribed, maximum inscribed and
minimum zone data fits.
Few attempts have been made by previous researchers to develop methods
for evaluating flatness error. The least-squares method (LSM) that minimizes
the sum of the squared deviations of the measured points from a fitted
feature has been suggested [21]. Although the least-squares techniques are
based on sound mathematical principles, the error values obtained are not
the minimum. The normal least-square fit has also been tried [22], but the
values obtained are not the minimum. To obtain the minimum zone solution,
numerical methods based on the Monte Carlo, Simplex and Spiral Search
echniques [23] have also been suggested. Li et al., [24] has suggested a new
simple approach called the convex-hull edge technique that gives the
minimum value of form error.
Flatness error
Flatness is one of the most common features in precision coordinate
metrology, and various criteria may be used for flatness error evaluation.
Flatness error is defined as the distance between two parallel planes that
contain the evaluated surface. Assuming 𝑃𝑖( 𝑥𝑖, 𝑦𝑖, 𝑧𝑖)( 𝑖 = 1,2, … , 𝑛) is the
measured points extracted by measuring a plane part. A flatness tolerance
specifies a tolerance zone defined by two parallel planes within which the
surface must lie. If all extracted data-points 𝑃𝑖( 𝑥𝑖, 𝑦𝑖, 𝑧𝑖) are between two
parallel planes, the minimum separation between these two parallel planes is
called the minimum zone solution (MZS) of flatness error (see ‫العثور‬ ‫يتم‬ ‫لم‬ !‫خطأ‬
.‫المرجع‬ ‫مصدر‬ ‫.)على‬ Assuming one of the two parallel plane equations of MZS is
[25]:
𝑧 = 𝑎𝑥 + 𝑏𝑦 + 𝑐
(3)
The distance 𝑑𝑖 from datapoints 𝑃𝑖( 𝑥𝑖, 𝑦𝑖, 𝑧𝑖) to the parallel plane is:
𝑑𝑖 =
𝑧 𝑖−𝑎𝑥 𝑖−𝑏𝑦 𝑖−𝑐
√1+𝑎2+𝑏2
(4)

44
The minimum separation 𝑓 between these two parallel planes is:
𝑓 = min(max( 𝑑𝑖) − min( 𝑑𝑖))
= min (𝑚𝑎𝑥 (
𝑧𝑖 − 𝑎𝑥𝑖−𝑏𝑦𝑖 − 𝑐
√1 + 𝑎2 + 𝑏2
) − min (
𝑧𝑖 − 𝑎𝑥𝑖−𝑏𝑦𝑖 − 𝑐
√1 + 𝑎2 + 𝑏2
))
= min (𝑚𝑎𝑥 (
𝑧 𝑖−𝑎𝑥 𝑖−𝑏𝑦 𝑖
√1+𝑎2+𝑏2
) − min (
𝑧 𝑖−𝑎𝑥 𝑖−𝑏𝑦 𝑖
√1+𝑎2+𝑏2
))
(5)
Obviously, the minimum separation 𝑓 is a function of( 𝑎, 𝑏). Consequently,
evaluating the minimum zone flatness error is translated into searching the
values of( 𝑎, 𝑏), so that the separation 𝑓( 𝑎, 𝑏) is the minimum and this
minimum value is just the flatness error. It is a non-linear optimisation
problem.
Figure 1: Flatness by minimum zone method.
Results and discussion
To validate the proposed scheme, a case study includes three differentr cases
is provided, and the obtained results are compared with other methods
presented in the literature. The PSO, convex hull, improved convex hull, and
Least squares are developed and employed to estimate the flatness error. The
measuremed data from the plane surface are given in Table 1 [26]. The
procedures were programmed in the MATLAB environment and the results
from different methods were demonstrated in Table 2. Deterministic
methods, although effecient at a small scale, become impractical in large-scale
problems. In such a case, nature-inspired optimisation algorithms are
necessary. As shown in the table, the comparison shows that the global
optimum solution of flatness evaluation problem using the CS and PSO can be
give the exact solution. The CS and PSO algorithms take about 100 iterations
d Flatness error
Parallel
planes
Cartesian coordinate

45
to find the minimum zone solution of the flatness error, and the calculation
results of flatness using the CS and PSO algorithm are 1.961161 μm and
1.960123 μm, respectifly.
Table 1: Comparison of the results calculated by different methods.
Case
study
No. of
points
Flatness tolerance
CS PSO
Improved
Convex
hull
Convex
hull
Least
squares
Case
study 01
15 1.960123 1.961161 1.972027 2.3774 2.3774
Case
study 02
25 0.15321 0.15485 0.155150 0.1756 0.1856
Case
study 03
25 0.002214 0.002227 0.002627 0.002817 0.00303
Conclusions
Robust optimisation tool for Coordinate Measuring Machines is becoming
ever more important because of current industrial demands for higher
productivity at increasing quality levels. In this work, a new intelligent
technique based on the use of the Cuckoo Search optimisation algorithm for
flatness error estimation is proposed. Extensive simulations using Matlab
environment and measured data in conjunction with deterministic and
nature-inspired optimisation algorithms have been carried out to verify the
and show the effectiveness of the proposed scheme. The proposed algorithm
has been validated and compared with Particle Swarm Optimisation , Convex
hull, Improved Convex hull, and Least squares. Simulations and comparison
show that the CS algorithm outperforms the PSO and other conventional
algorithms, which can act as an alternative optimisation algorithm for CMM
flatness error software that can be used for quality control. It can therefore be
concluded that it is possible to optimise a flatness error using the Cuckoo
Search algorithm, which can be used to determining the acceptability of
manufactured parts. Future studies will concentrate on applications in other
automated manufacturing systems under different operation environments.

46
References
[1] J. Ni, "CNC machine accuracy enhancement through real-time error
compensation," Journal of manufacturing science and engineering, vol. 119,
pp. 717-725, 1997.
[2] E. Abele, Y. Altintas, and C. Brecher, "Machine tool spindle units," CIRP
Annals - Manufacturing Technology, vol. 59, pp. 781-802, 2010.
[3] W. Grzesik, "Experimental investigation of the cutting temperature
when turning with coated indexable inserts," International Journal of Machine
Tools and Manufacture, vol. 39, pp. 355-369, 1999.
[4] X.-S. Yang, Nature-inspired metaheuristic algorithms: Luniver press,
2010.
[5] M. Ye, X. Wang, and Y. Xu, "Parameter identification for proton
exchange membrane fuel cell model using particle swarm optimization,"
International Journal of Hydrogen Energy, vol. 34, pp. 981-989, 1// 2009.
[6] R. C. Eberhart and J. Kennedy, "A new optimizer using particle swarm
theory," in Proceedings of the sixth international symposium on micro machine
and human science, 1995, pp. 39-43.
[7] A. Abdulshahed, A. P. Longstaff, and S. Fletcher, "A particle swarm
optimisation-based Grey prediction model for thermal error compensation on
CNC machine tools," in Laser Metrology and Machine Performance XI,
LAMDAMAP 2015, Huddersfield, 2015, pp. 369-378.
[8] S. Liang, M. Rajora, X. Liu, C. Yue, P. Zou, and L. Wang, "Intelligent
Manufacturing Systems: A Review," International Journal of Mechanical
Engineering and Robotics Research, vol. 7, pp. 324-330, 2018.
[9] J. Freeman, A. White, and D. G. Ford, "Ball-screw thermal errors-a finite
element simulation for on-line estimation," in Laser Metrology and Machine
Performance V, Southampton, 2001, pp. 269-278.
[10] S. Fletcher and D. G. Ford, "Measuring and modelling heat transfer and
thermal errors on a ballscrew feed drive system," in Laser Metrology and
Machine Performance VI, 2003, pp. 349-360.
[11] P. B. Fernandes, R. C. L. De Oliveira, and J. V. F. Neto, "A Modified QPSO
for Robotic Vehicle Path Planning," in 2018 IEEE Congress on Evolutionary
Computation (CEC), 2018, pp. 1-7.

47
[12] A. R. Yildiz, "Cuckoo search algorithm for the selection of optimal
machining parameters in milling operations," The International Journal of
Advanced Manufacturing Technology, vol. 64, pp. 55-61, 2013.
[13] A. H. Gandomi, X.-S. Yang, and A. H. Alavi, "Cuckoo search algorithm: a
metaheuristic approach to solve structural optimization problems,"
Engineering with computers, vol. 29, pp. 17-35, 2013.
[14] J. Huang, L. Gao, and X. Li, "An effective teaching-learning-based
cuckoo search algorithm for parameter optimization problems in structure
designing and machining processes," Applied Soft Computing, vol. 36, pp. 349-
356, 2015.
[15] W. C. E. Lim, G. Kanagaraj, and S. Ponnambalam, "PCB drill path
optimization by combinatorial cuckoo search algorithm," The Scientific World
Journal, vol. 2014, 2014.
[16] W. C. E. Lim, G. Kanagaraj, and S. Ponnambalam, "Cuckoo search
algorithm for optimization of sequence in pcb holes drilling process," in
Emerging trends in science, engineering and technology, ed: Springer, 2012,
pp. 207-216.
[17] P. Civicioglu and E. Besdok, "A conceptual comparison of the Cuckoo-
search, particle swarm optimization, differential evolution and artificial bee
colony algorithms," Artificial Intelligence Review, vol. 39, pp. 315-346, 2013.
[18] X.-S. Yang and S. Deb, "Cuckoo search via Lévy flights," in Nature &
Biologically Inspired Computing, 2009. NaBIC 2009. World Congress on, 2009,
pp. 210-214.
[19] X.-S. Yang, Cuckoo search and firefly algorithm vol. 37: Springer, 2014.
[20] I. Fister Jr, X.-S. Yang, I. Fister, J. Brest, and D. Fister, "A brief review of
nature-inspired algorithms for optimization," arXiv preprint arXiv:1307.4186,
2013.
[21] G. Samuel and M. Shunmugam, "Evaluation of circularity and sphericity
from coordinate measurement data," Journal of materials processing
technology, vol. 139, pp. 90-95, 2003.
[22] G. Samuel and M. Shunmugam, "Evaluation of straightness and flatness
error using computational geometric techniques," Computer-Aided Design,
vol. 31, pp. 829-843, 1999.

48
[23] G. Samuel and M. Shunmugam, "Evaluation of circularity from
coordinate and form data using computational geometric techniques,"
Precision engineering, vol. 24, pp. 251-263, 2000.
[24] P. Li, X.-M. Ding, J.-B. Tan, and J.-W. Cui, "A hybrid method based on
reduced constraint region and convex-hull edge for flatness error evaluation,"
Precision engineering, vol. 45, pp. 168-175, 2016.
[25] X.-L. Wen, X.-C. Zhu, Y.-B. Zhao, D.-X. Wang, and F.-L. Wang, "Flatness
error evaluation and verification based on new generation geometrical
product specification (GPS)," Precision Engineering, vol. 36, pp. 70-76, 2012.
[26] X. Zhu and H. Ding, "Flatness tolerance evaluation: an approximate
minimum zone solution," Computer-Aided Design, vol. 34, pp. 655-664, 2002.

49
Appindeces
Table 2: The measurement data from the plane surface.
No x (μm) y (μm) z (μm)
1 -2 1 5
2 -1 1 4
3 0 1 1
4 1 1 2
5 2 1 2
6 -2 0 4
7 -1 0 3
8 0 0 3
9 1 0 2
10 2 0 2
11 -2 -1 3
12 -1 -1 4
13 0 -1 2
14 1 -1 1
15 2 -1 2

50
1 0.2 0.2 -0.0664
2 0.2 0.4 -0.0644
3 0.2 0.6 0.0088
4 0.2 0.8 -0.0112
5 0.2 1 -0.0624
6 0.4 0.2 -0.0383
7 0.4 0.4 0.0655
8 0.4 0.6 0.0636
9 0.4 0.8 0.0285
10 0.4 1 -0.0061
11 0.6 0.2 -0.0952
12 0.6 0.4 -0.0115
13 0.6 0.6 -0.0241
14 0.6 0.8 0.0352
15 0.6 1 -0.02
16 0.8 0.2 0.0154
17 0.8 0.4 -0.0132
18 0.8 0.6 -0.0222
19 0.8 0.8 0.0771
20 0.8 1 -0.0004
21 1 0.2 0.0577
22 1 0.4 -0.0562
23 1 0.6 0.0921
24 1 0.8 0.0654
25 1 1 -0.0212

51
1 0.2556 0.2994 0.0005
2 1.4992 0.3371 0.0013
3 2.6656 0.3726 0
4 3.5978 0.4009 0.0005
5 4.6241 0.4321 -
0.0007
6 4.5989 1.264 0.0001
7 3.4451 1.2289 0.0008
8 2.7096 1.2066 0.0004
9 1.6726 1.2968 0.0014
10 0.5273 1.262 0.0009
11 0.1683 2.1413 -
0.0002
12 0.9906 2.1663 0.001
13 2.5485 2.1801 0.0008
14 3.4605 2.1369 0.0011
15 4.8632 2.1795 -
0.0017
16 4.8401 2.9417 -
0.0014
17 3.6557 2.9058 0.0012
18 2.4224 2.8683 0.0012
19 1.3839 2.8368 0.0011
20 0.4966 2.8098 -
0.0002
21 0.4672 3.7751 -
0.0008
22 1.6709 3.8116 0.001
23 2.8864 3.8486 0.0006
24 3.7562 3.875 0.0008
25 4.6746 3.9029 -
0.0003
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Efficient evaluation of flatness error from Coordinate Measurement Data using Cuckoo Search optimization algorithm

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Efficient evaluation of flatness error from Coordinate Measurement Data using Cuckoo Search optimization algorithm