Scientific Programming

Volume 2018, Article ID 2483781, 13 pages

https://doi.org/10.1155/2018/2483781

## Cylindricity Error Evaluation Based on an Improved Harmony Search Algorithm

^{1}Key Laboratory of Intelligent Manufacturing and Robotics, School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China^{2}Department of Mechanical Engineering, Hubei University of Automotive Technology, Shiyan 442002, China

Correspondence should be addressed to Yang Yang; nc.ude.uhs.i@ifyyrm

Received 30 March 2018; Revised 7 June 2018; Accepted 14 June 2018; Published 19 July 2018

Academic Editor: Ricardo Soto

Copyright © 2018 Yang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The cylindricity error is one of the basic form errors in mechanical parts, which greatly influences the assembly accuracy and service life of relevant parts. For the minimum zone method (MZM) in international standards, there is no specific formula to calculate the cylindricity error. Therefore, the evaluation methods of the cylindricity error under the MZM have been widely concerned by international scholars. To improve the evaluation accuracy and accelerate the iteration speed of the cylindricity, an improved harmony search (IHS) algorithm is proposed and applied to compute the cylindricity. On the basis of the standard harmony search algorithm, the logistic chaotic initialization is introduced into the generation of initial solution to improve the quality of solutions. During the iterative process, the global and local search capabilities are balanced by adopting the *par* and operators adaptively. After each iteration, the Cauchy mutation strategy is adopted to the best solution to further improve the calculation precision of the IHS algorithm. Finally, four test functions and three groups of cylindricity error examples were applied to validity verification of the IHS algorithm, the simulation test results show that the IHS algorithm has advantages of the computing accuracy and iteration speed compared with other traditional algorithms, and it is very effective for the application in the evaluation of the cylindricity error.

#### 1. Introduction

In order to evaluate the manufacturing quality of mechanical parts, the inspection process has always been an important step in the whole life cycle of mechanical products. As an important parameter to evaluate the manufacturing accuracy of the internal and external cylindrical surfaces of the rotary part, the cylindricity error has become the focus of the research in the field of mechanical measurement. Coordinate measuring machine (CMM) is a common tool to inspect the cylindricity error of the mechanical parts, which is widely used because of its high precision. The spatial coordinates information of the measured point of the part is obtained by the sensor of the CMM. Then, the measurement data obtained from the sensor is analyzed through the error evaluation algorithm to calculate the cylindricity error of the measured parts. At present, in order to obtain the cylindricity error of measurement data more quickly and accurately, a lot of research results have been obtained. It is of great value in engineering application to evaluate the cylindricity error quickly and accurately, which has great influence on the function and life of the product.

For the cylindricity error evaluation, the least square method (LSM) is widely used in the field of engineering applications, but the accuracy of LSM is relatively low and cannot be applied in the high-precision evaluation field [1, 2]. Compared with the LSM, the minimum zone method (MZM) is one of the techniques that has a higher evaluation accuracy. It can meet the minimum zone principle in the international standards and obtain the accurate results of the measurement data. However, there is no specific algorithm in international standards, so it has received widely attention and research on the MZM by international scholars.

Based on the initial research results, the nonlinear optimization algorithms [3, 4] are the most important methods for cylindricity error evaluation under the MZM. Moreover, the nonlinear optimization algorithms are relatively complex and difficult in the construction of the mathematical model. However, with the continuous improvement of the artificial intelligence technology, the intelligent optimization algorithms have developed an effective method to solve such complex nonlinear problems. And some of them have been introduced into the cylindricity error evaluation because of their simplicity and effectiveness, such as GA [5] and PSO [6, 7]. In addition, compared with nonlinear optimization algorithms, the construction of the mathematical model for intelligent optimization algorithm is also very simple. But the initialization parameters of these algorithms have great influence on the calculation results. Therefore, the improvement of the accuracy and convergence speed of the cylindricity algorithm has become an important research topic at this stage.

Harmony search (HS) algorithm is an intelligent optimization algorithm, which simulates the creation process of the music [8]. It has the advantages of relatively simple computational principle and strong global search ability, so it is very suitable for engineering applications such as the sizing optimization of truss structures and reliability problems [9, 10]. But for the basic HS algorithm, there are some problems such as the premature convergence in the iteration process and low accuracy in later iteration.

To improve the evaluation accuracy and accelerate the iteration speed of the cylindricity, on the basis of the standard HS algorithm, the logistic chaotic initialization is introduced into the generation of initial solution to improve the quality of solutions. During the iterative process, the global and local search capabilities are balanced by adopting the *par* and operators adaptively. After each iteration, the Cauchy mutation strategy is adopted to the best solution to further improve the calculation precision of the IHS algorithm. Finally, four test functions and three groups of cylindricity error examples were applied to validity verification of the IHS algorithm, the simulation test results show that the IHS algorithm has advantages of the computing accuracy and iteration speed compared with other traditional algorithms, and it is very effective for the application in the evaluation of the cylindricity error.

The organizational structure of this paper is shown as follows: the research background and significance of the paper are introduced in Section 1. In Section 2, the present research results of this work are described. In Section 3, the mathematical model of the cylindricity error and the objective function of the problem are established. The basic HS algorithm and IHS algorithm are proposed in Sections 4 and 5, respectively. The simulation experiments of the algorithms and cylindricity error analysis are carried out in Section 6. In the last section of this paper, the conclusion of this work is summarized and the future work is prospected.

#### 2. Related Works

In the research results of the cylindricity error evaluation under the MZM, Carr and Ferreira established the nonlinear mathematical model of the cylindricity and straightness errors and then used the linear programming method (LPM) to solve the problem, but this method requires a set of appropriate initial solutions [3]. Lai and Chen converted the spatial cylindricity into the plane problems by nonlinear transformation and used the parameters adjustment to obtain the cylindricity error [4]. Chou and Sun deduced a general mathematical model of the cylindricity error. The simulated annealing (SA) algorithm is applied to solve the model that shows the advantage in the global optimization performance [11]. Lai et al. used the genetic algorithm (GA) to solve the mathematical model of the cylindricity error under the MZM, and the experiment proves that the method has good flexibility and accuraxcy [5]. Weber et al. applied the linear approximation technique to the form error evaluation that includes the cylindricity error. The method provided an approximate solution for the form error [12]. Zhu and Ding combined the motion geometry and sequence approximation algorithm to analyze and evaluate the cylindricity error. The experiments proved the validity of the algorithm [13]. Lao et al. improved the hyperboloid technique by constructing an initial axis to establish the cylindrical error evaluation model, and the experiment showed that the method further improved the accuracy of cylindricity evaluation [14]. Cui et al. and Mao et al. applied a particle swarm optimization (PSO) algorithm to cylindricity error evaluation, and the calculation accuracy was further improved, respectively [6, 7]. Venkaiah and Shunmugam designed a cylindricity error evaluation algorithm based on the computational geometry method [15]. Bei et al. and Guo et al. improved the genetic algorithm and applied it to the cylindricity error assessment, respectively [16, 17]. Li et al. used the coordinate transformation to construct the cylindricity error mathematical model, and the model has a high evaluation accuracy [18]. Luo et al. further improved the global optimization ability of an artificial bee colony algorithm by introducing tabu strategy, thus increasing the evaluation speed of the cylindricity error [19]. Wen et al. modified the particle velocity using a contraction factor and improved the convergence ability of the PSO. The improved PSO (IPSO) was applied to the cylindricity and conicity errors which have a good flexibility [20]. Lei et al. designed a method of cylindricity error evaluation based on the geometry optimization searching algorithm. The method can obtain the cylindricity error by setting the hexagon in the measurement points [21]. Lee et al. developed a cylindricity error evaluation based on support vector machines (SVM), and the algorithm can be applied in the field of the machine vision system [22]. Wen et al. utilized a quasiparticle swarm algorithm and calculated the cylindricity error [23]. He et al. used kinematic geometry and a sequential quadratic programming algorithm to solve the cylindricity error of measured parts. The method has high stability and accuracy [24]. Hermann introduced the application of the computation geometry technique to the form errors that include cylindricity [25]. Zheng et al. adopted a linear programming method to construct the mathematical model of the cylindricity error. The model was solved by a new simplex method [26].

On the basis of the above literatures, we improved the HS algorithm through the generation of initial solution, the dynamic adjustment of operators, and the optimal solution perturbation, to further enhance the optimization ability and the iterative speed of the basic HS algorithm. It is designed to provide a reliable method for the evaluation of the cylindricity error.

#### 3. Mathematical Model of Cylindricity Error Evaluation

According to the concepts in the relevant standard [1, 2], the cylindricity error under the MZM can be defined as follows: the area is the radius difference of two coaxial ideal cylinders that contain the measured cylinder, and when the area reaches the minimum value, the radius difference is the cylindricity error under the MZM. As shown in Figure 1, the graph shows the measurement process of the cylindricity error. For the cylindrical parts, due to the manufacturing errors of the machine tool, the actual parts will always have a certain error compared with the ideal parts, and the cylindrical surface will produce a certain deformation. Therefore, the CMM is used to obtain the coordinates information of the measured parts. Figure 2 illustrates the principle of the cylindricity error calculation, where is the axis of the two ideal cylinders and is the radius difference; when takes the minimum value, the value of is the cylindricity error of the cylinder measured under the MZM.