Minimum Zone Circle Research Articles

A new method for evaluating roundness error based on improved bat algorithm

Roundness error is one of the core indicators for evaluating the geometric accuracy of round parts mechanical products, directly affecting product performance and service life. In the field of metrology, the accurate, rapid and standardized assessment of roundness error has always been a hot topic. This paper proposed a new method for evaluating roundness error based on improved bat algorithm. This method was based on the new geometric product specification for extraction, filtering and fitting. Based on the advantages of solving optimization problems using the bat algorithm, transformed the roundness error evaluation problem using the minimum zone method into a problem of using the bat algorithm to find the center of the minimum zone circle, then further solved for the roundness error value. Also, this algorithm effectively avoided falling into local optimal solutions by introducing chaotic inertia weights during the velocity update phase, it improved the accuracy and speed of the evaluation. Introduced adaptive parameters during the loudness and emission rate update phases to enhance the algorithm’s global search capability, it enhanced the stability of the algorithm. The experimental results indicated, the efficiency of roundness error evaluation in the new method was significantly better than that of the genetic algorithm, the simplex algorithm and the emperor penguin algorithm. There was a significant improvement in evaluation accuracy and stability. The feasibility of this method in roundness error evaluation using the minimal zone method was validated.

A novel roundness error evaluation method for high-speed EMU train axles

This article proposes a novel roundness error evaluation method for high-speed electric multiple units (EMU) train axles to evaluate the roundness error of the minimum zone circle. This method utilizes the adaptive ability of Bayesian linear regression to the data to solve the initial center of the circle, avoiding the multiple search process, which not only improves the data utilization and simplifies the solving steps but also has a strong noise immunity. The method first establishes a Hough space model, maps the collected data points to the Hough space, and forms different conical surfaces; then, by Bayesian linear regression, the probability density function is set with the distance from the space point to the conical surface as a parameter, and through continuous iteration, the point with the highest probability value is the more accurate initial center of the circle. Finally, the final optimal center of the circle is obtained by the principle of minimum inclusion region. We conducted experimental studies on axles of different diameters and compared them with other methods. The results prove the practicality and effectiveness of the technique.

A fast hybrid roundness evaluation algorithm based on computational geometry and particle swarm optimization for profiles with massive points

The advancement of measurement has led to rigorous demands for the accuracy and efficiency of roundness evaluation of minimum zone circle (MZC) for profile with massive points. Existing algorithms struggle to balance efficiency and accuracy in such cases. A fast hybrid roundness evaluation (HCGPSO) algorithm for MZC based on the computational geometry (CG) and particle swarm optimization (PSO) algorithm is proposed, which incorporates a CG-based global best (gBest) selection method into the PSO algorithm and refines the termination criterion. A comparison with the published studies demonstrated that the HCGPSO algorithm were accurate and result are displayed to 10-15 mm. The impact of target accuracy, the harmonic and the number of points contained in profile on the performance of proposed algorithm are analyzed. With sub-nanometer target evaluation precision, HCGPSO enhances efficiency by an average of 64 % for profiles containing the highest harmonics from 30 undulations per revolution (upr) to 500 upr with 1500 points. For profiles with 300 to 4000 points at 150 upr, an average efficiency enhancement of 60 % was observed. For profiles with 4000 points at 500 upr from four workpieces, the calculation time is reduced by 55 % on average compared to the PSO algorithm, with increased accuracy and stability.

A Novel Iterative Field Search Approach to Minimum Zone Circle for Roundness Error Estimation

Abstract Roundness is one of the common attributes in the manufacturing industry. Roundness is the most prominent of the extant fundamental forms, as the majority of the fabricated components are round or cylindrical. The examination of roundness error associated with such features is critical since inadequate evaluation might result in the rejection of excellent parts. The performance of any equipment depends on the mating parts. Out-of-roundness components cause the inefficiency of such equipment’s system performance. As a result, roundness error assessment is critical for macro-sized specimens as well as for micro- and nano-sized components. The present article proposes a novel approach to Minimum Zone Circle (MZC) to evaluate the roundness error. An iterative field search methodology is used in the suggested approach. The proposed methodology evaluates the roundness error based on the generated discretized points within the search space and continues the same by decreasing the search space with an increase in the number of iterations to attain the Minimum Zone Error (MZE). The proposed algorithm has been tested with ten CMM (coordinate measuring machine) datasets and ten form datasets available in the literature studies and found to be excellent in comparison to the existing techniques. Further, the proposed methodology was also implemented to estimate the MZE of centerless ground specimens at multiple cross sections, and the roundness error obtained was lesser as compared to the LSC, MCC, and MIC. The number of generated discretized points is flexible and can be varied to reduce the number of iterations and computations. The recommended method is quite effective in assessing the roundness error when compared to the existing techniques. It also works well on data that was both evenly and unevenly spaced. The results suggest that altering the search field area for higher computing efficiency is straightforward, resilient, and versatile.

Online measurement method for roundness of large aluminum alloy ring based on laser ranging principle

Aluminum alloy rings are widely used in aerospace, energy, petrochemical, and equipment applications. The loss of roundness often occurs because of the weak rigidity and unstable plasticity of aluminum alloy rings. Therefore, an online measurement technique for the roundness of aluminum alloy rings is critical for ensuring the processing quality of aluminum alloy rings. In this paper, an online roundness measurement method based on the laser ranging principle and improved particle swarm optimization (IPSO) is proposed with the aid of the rotation of the ring during the rolling process. First, the distance from the outer surface of the ring to the laser rangefinder was measured using a triangular laser rangefinder with the help of the rotation of the ring in the rolling process. Second, the distance data were denoized using an improved Kalman filter algorithm to reduce the influence of random noise on the measurement results. Finally, the roundness and geometric center were evaluated using the IPSO algorithm combined with the minimum zone circle (MZC) method and compared with the results obtained using the standard particle swarm optimization (PSO) algorithm, genetic algorithm (GA), and least square (LS) algorithm. The experimental results show that the roundness measured by the IPSO algorithm is less than 0.06 mm, demonstrating better performance in the roundness evaluation. The measurement method for roundness proposed in this paper is suitable for the online evaluation of large aluminum alloy rings, avoiding the influence of ring size on roundness measurement.

New accurate algorithms of circularity evaluation

The minimum circumscribed circle (MCC), maximum inscribed circle (MIC), and minimum zone circle (MZC) methods for circularity evaluation are difficult to execute due to the lack of specific rules in mathematics, especially the MZC. New accurate algorithms have been proposed to realize the MIC, MCC and MZC evaluation methods. First, the diameter criterion and acute triangle criterion of control points defining the MIC or MCC are presented. The definition of the crossing sector structure is introduced in the minimum zone criterion and transformed into an angular relationship of control points, making it easy to identify the MZC. Then the key points of the algorithms are presented in details including the initial circle or the initial region, centre moving direction, minimum step size and updating rules of control points. Flow charts are provided to make the algorithm producible. Finally, four cases are selected to demonstrate the advantages of the algorithms. The results show that the proposed MZC algorithm is more general than that in the published reference. Compared to the traditional ergodic searching algorithms, the efficiency of the new algorithm for the MIC, the MCC and the MZC evaluations of circularity can be improved by 100% when the point set increases to 100 and above.

Roundness Holes’ Measurement for milled workpiece using machine vision inspection system based on IoT structure: A case study

Manufacturing Metrology is a vital aspect of the sophisticated industrial 4.0-based manufacturing industry, as practically everything machined with certain metrological systems has to be measured or inspected. The Internet of Things (IoT) is a critical technology for industry 4.0 development. Furthermore, modern industrial devices should communicate via local (edge) and cloud computing servers. The Message Queuing Telemetry Transport (MQTT) protocol intends to increase the performance for sending/receiving roundness holes’ data measuring from /to the server-based cloud. An error in its roundness mainly involves the inspection of circular portions. Roundness errors generate assemblies, lubricant distribution, and rotational force creation challenges. This study proposed applying the 3SMVI system and validating the system's by utilising the minimum zone circle tolerance algorithm to provide the precise error value and determine the roundness circle's error between the five circles. Hence, the approach is currently used for measuring the roundness circle error in the camera industry throughout the manufacture of components. The experiment was conducted using the smart system based on interpreted STEP-NC files for machine vision inspection (3SMVI) for non-contact and Coordinate Measuring Machine (CMM) for contact approaches by comparing the results. The Contact Measurement Method (CMM) is one of the most precise and time-consuming ways to inspect a round object for imperfections. The article validates the development of the 3SMVI system based on IoT architecture by utilising image processing technologies to measure errors in the circular roundness that can overcome these difficulties. The findings of both approaches showed a difference between 4.6 µm and 9.6 µm. However, Precession can be enhanced by a high-resolution camera with the right light.

Improved algorithm for minimum zone of roundness error evaluation using alternating exchange approach

Based on computational geometry techniques, an improved algorithm for the minimum zone of roundness error evaluation using an alternating exchange method is presented. A minimum zone fitting function was created to enhance the roundness error evaluation. The function uses three candidate points to determine the initial solution: the expected centre, the mean circle radius, and the corresponding zone half-width. The best solution function is designed to use the initial solution as the input to determine the optimum solution for the minimum zone circle (MZC). The proposed algorithm was validated using data available in the literature. The roundness error evaluation comparison results demonstrate that the proposed method accurately detects both the centre error magnitude and MZC and overcomes the insufficiency of using selected colinear points for four selected points.

Improved evaluation of minimum zone roundness using an optimal solution guidance algorithm

This study proposes a high speed and accurate evaluation method for minimum zone roundness (MZR) using an optimal solution guidance algorithm (OSGA). The least-square center and roundness determine an original search zone, then the initial candidate points are configured as the centers of a minimum zone circle (MZC). To enhance the global search ability, subsequent candidate points follow the current optimal candidate points. The best three current candidate points jointly generate the new candidate points for the next iteration. In addition, a dynamic search zone updating strategy is proposed to achieve fast convergence of the optimal solution. The width of the search zone varies with the number of iterations required to achieve this rapid convergence. The effectiveness and performance of the proposed method are validated using datasets from previous studies. The proposed OSGA achieves excellent convergence performance, and reaches the global optimal solutions more quickly than previous methods. Across four different datasets, the termination condition is satisfied in only 0.0031–0.0058 s, the optimal solutions are obtained within 5–15 iterations, and the standard deviation of the MZR is less than 5.8276 × 10‒6 mm. The proposed method is also expected to extend to rapid and high-precision evaluation of cylindricity or straightness.

A Novel Iterative-Based Field Search Technique for Roundness Evaluation

Circularity is one of the frequently observed features in manufacturing industry. Most of the components manufactured are unions of the fundamental shapes, one of which is the circle. Due to very close tolerances, micro and nano-sized components’ manufacturing serves a very challenging task. Slight variance in the manufactured component can lead to incorrect assembly and inefficient system performance. Hence, the roundness evaluation of any circular component is important. The present article proposes a novel algorithm to evaluate the roundness error based on the minimum zone circle (MZC) concept. The proposed algorithm is an iterative field search technique. Compared to the existing approaches, the proposed method is very effective in determining the roundness error. Furthermore, it also performed well both on uniformly and non-uniformly spaced data. The results show that it is simple, robust and flexible enough to vary the search field area for better computational efficiency.

A simple method for estimating the roundness of minimum zone circle

AbstractWith the rapid development of micromachining technology, the feature size of object parts becomes smaller whilst the required tolerance accuracy becomes higher. The geometry of these micro‐parts must be critically verified at sub‐micrometer or even nanometer accuracy in three dimensions to guarantee machining quality, such as the cylindrical parts. Therefore, the method for estimating roundness error plays an increasingly important role in machining. This paper focuses on the roundness estimation based on the minimum zone circle method in a Cartesian coordinate system. An asymptotic search method is also proposed to obtain the coordinates of the concentric centre of the minimum zone circle model and to calculate the roundness error. Comparison and simulation experiments are also conducted to test the performance of the proposed method. The results demonstrate that the proposed method is effective, reliable and can meet the requirements for roundness estimation.

Improved evaluation of minimum zone roundness by integrating bidirectional search of unequal probability and offset mechanisms

This paper reports an improved method that integrates a bidirectional search of unequal probability and an offset movement mechanism (BSOM) to evaluate the minimum zone roundness (MZR) with high speed and accuracy. The centre and roundness of a least-squares circle are first solved to construct an original search zone, and the initial candidate points of the minimum zone circle centre are configured. A bidirectional search of unequal probability that considers the subsequent points either toward or away from the current best candidate point is designed to enhance the global search ability. Considering the candidate points at different located positions have different effects on convergence characteristics, the search area is divided into two sections based on the average distance between the current best candidate point and the other candidate points, and different forward and backward search probabilities are configured. An adaptive offset movement mechanism is integrated to generate new candidate points for the next generation. The effectiveness of the proposed method is experimentally verified using datasets from previous studies. The configurations of BSOM parameters and influence of the distribution and number of data points on evaluation accuracy and efficiency are analyzed. The experimental examples and comparison results demonstrate the excellent convergence performance of the proposed method, which reaches the global optimum faster than previous methods. Across six different datasets, only 0.0095 s–0.0282 s is required when the average distance D ⩽ 1.0 × 10−7 mm as the iteration convergence condition, and the standard deviation of the MZR is less than 5.3777 × 10−9 mm, while the search parameters are set as P1 = 0.6, P2 = 0.95, and b = 0.6. The proposed approach is expected to be suitable for on-line evaluation of roundness error.

Comparative study of roundness evaluation algorithms for coordinate measurement and form data

Roundness error is one of the significant parameters used in quality analysis and control of circular or cylindrical components. In both laboratories and industries, direct estimation of roundness error is predominantly obtained using the coordinate measuring machine and form testing device. The literature is copiously supplied with different approaches/algorithms for the implementation of the reference circles that are used to quantify roundness error. A test for their relative performance is of relevance in order to make a decision about which approach is to be used for the execution of a given reference circle. This paper replicates and compares a few selected benchmark algorithms from literature to make recommendations for the optimum choice of execution methods for Minimum Circumscribed Circle (MCC), Maximum Inscribed Circle (MIC) and Minimum Zone Circle (MZC) that are used to compute roundness error. A new computational geometric concept using reflection mapping technique was formulated to assess roundness using MIC and MZC, which has also been used in the comparison. All the algorithms were tested using the same ten sets of coordinate/profile data to establish their relative efficacy in roundness error computation. It was found that no single computation concept consistently gave the best results for all the three reference circles simultaneously. Thus, the three reference circles have been independently analyzed, and recommendations for the algorithm to be used for each of them have been presented individually.

Evaluation of roundness error using a new method based on a small displacement screw

In relation to industrial need and the progress of technology, LNE would like to improve the measurement of its primary pressure, spherical and flick standards. The spherical and flick standards are respectively used to calibrate the spindle motion error and the probe which equips commercial conventional cylindricity measuring machines. The primary pressure standards are obtained using pressure balances equipped with rotary pistons with an uncertainty of 5 nm for a piston diameter of 10 mm. Conventional machines are not able to reach such an uncertainty level. That is why the development of a new machine is necessary. To ensure such a level of uncertainty, both stability and performance of the machine are not sufficient, and the data processing should also be done with accuracy less than a nanometre. In this paper, a new method based on the small displacement screw (SDS) model is proposed. A first validation of this method is proposed on a theoretical dataset published by the European Community Bureau of Reference (BCR) in report no 3327. Then, an experiment is prepared in order to validate the new method on real datasets. Specific environment conditions are taken into account and many precautions are considered. The new method is applied to analyse the least-squares circle, minimum zone circle, maximum inscribed circle and minimum circumscribed circle. The results are compared to those done by the reference Chebyshev best-fit method and reveal perfect agreement. The sensibilities of the SDS and Chebyshev methodologies are investigated, and it is revealed that results remain unchanged when the value of the diameter exceeds 700 times the form error.

Applying particle swarm optimization algorithm to roundness error evaluation based on minimum zone circle

Minimum zone circle (MZC) method and least square circle (LSC) method are two most commonly used methods to evaluate roundness, but only the MZC method complies with the standard definition and can obtain the minimum roundness error value. The determination of the center of MZC is a nonlinear optimization problem which is suitable to be solved by particle swarm optimization (PSO) algorithms. In this paper, the standard PSO algorithm was introduced and theory analysis about the impact of value selection of some important parameters, such as inertia weight ω, on the algorithm’s stability and convergence was carried on so as to provide basis for giving these parameters better values. Furthermore, the superiority of making ω decrease linearly with iterations was verified through a computation experiment in terms of stability and accuracy, compared with the other three cases of ω=1, 0.5, 0. Based on the analysis, the novel PSO algorithm, with ω decreasing linearly from 0.9 to 0.4 and the LSC center as the initial positions of the particles, is implemented to obtain MZC-based roundness errors of sampling points collected from circular section profiles by a coordinate measuring machine (CMM). By comparing the novel PSO–MZC results with the LSC-based results, it is concluded that the former are a little smaller than the latter, which verifies that the novel PSO algorithm is feasible to calculate roundness error and the fact that a LSC-based one is generally larger than a MZC-based result; the values of the two roundness errors are both related to sample size and increase with an increase in the sample size with a decreasing increment.

Determination of the minimum zone circle based on the minimum circumscribed circle

An approach for solving the minimum zone circle based on the minimum circumscribed circle is presented in this note. Three candidate points of the minimum zone circle are selected according to the minimum circumscribed circle. Another candidate point can be determined based on the distance operator easily. It has been carried out to solve the minimum zone circle effectively.

Minimum Zone Evaluation for Roundness Error Based on Geometric Approximating Searching Algorithm

Considering the characteristics of roundness error, a new method for roundness error evaluation in the rectangular coordinates, named as Geometric Approximation Searching Algorithm, is presented in this paper. The principle and steps of the algorithm are described in detail. The mathematical formulas and program flowchart are given. At first the algorithm allocates a square and predetermines the ideal centre point as the initial reference point. The radius value of all the measured points are calculated by each corresponding vertexes of the square respectively. After each vertex of the square and the initial reference point are used as the ideal centre point to calculate the roundness error, the minimum difference of the radius is obtained. The judgment and arranged square can be done repeatedly. Finally, the roundness error value of the minimum zone circle is determined. The experimental results show that the roundness error can be evaluated effectively and accurately by using this algorithm.

Metrological characterization of interior circular features using digital optical machines: Calculation models and application scope

The objective of this work is to analyze and compare common methods used for indirect metrological characterization of circular geometrical features: least square circle, minimum zone circle, maximum inscribed circle and minimum circumscribing circle, based on images obtained using digital optical machines. The self-developed calculation algorithms used were implemented with Matlab® software considering the following study variables: the amplitude of the angular sector of the circular feature, its nominal radius and the magnification used by the optical machine. Under different measuring conditions the radius and roundness error of an angular sector of a piece of LEGO and of a ring gauge were determined. That is, the proposed algorithms allow the characterization of both full circumferences as angular sectors. We were able to determine the degree of precision of each method and its scope of application by comparing the results obtained with different adjustments to values certified by an accredited laboratory.

Development and application of α-hull and Voronoi diagrams in the assessment of roundness error

Computational geometry has been used to select effective data points from the measured data points for evaluating the roundness error to improve the computational complexity. However, for precision parts most of the measured points are on the vertices of the convex hull; it cannot have any effect on improving the computational complexity with the Voronoi diagrams. In this paper the roundness error is evaluated with α-hull and the Voronoi diagram instead of convex hull. An approach for constructing α-hull with the minimum radius separation is presented to determine the vertices of the Voronoi diagram. The experimental results showed that the roundness error of the minimum zone circle could be solved efficiently with α-hull and the Voronoi diagram.

An Improved Particle Swarm Optimization Algorithm Applied to the Unified Evaluation of Circularity Error

Minimum zone circle (MZC), minimum circumscribed circle (MCC), maximum inscribed circle (MIC) and least square circle (LSC) are four common methods used to evaluate circularity errors. A novel particle swarm optimization algorithm based on self-adapted comprehensive learning (ACL-PSO) is proposed to evaluate circularity errors with real coded strategy. In the algorithm, population learning mechanism and velocity mutation strategy are adopted. In the meantime, ACL-PSO is applied to the unified evaluation of circularity error. The experiment results evaluated by different methods indicate that the proposed algorithm not only converges to the global optimum rapidly, but also has good stability, and it is easy to generalize.