Minimum Zone Circle Research Articles

Method for Roundness Error Evaluation Based on Geometry Optimization

According to the characteristics of roundness error,a new algorithm of evaluating roundness error based on geometry optimization is presented.The measured circle sampling points in rectangular spatial coordinates and the minimum zone circle (MZC),the minimum circumscribed circle (MCC) and the maximum inscribed circle(MIC) errors can be evaluated simultaneously in the error evaluating method.The principle and step of using the algorithm to solve the roundness error is described in detail and the mathematical formula and program flowchart are given.The optimization method and linearization method and uniform sampling are not adopted in the algorithm.This algorithm need not satisfy assumed small error or small deviation and only calls the formula of distance between point to point repeatedly.The principle of the algorithm is that a hexagon is collocated on the basis of the initial reference point,the radius value of all the measured points are calculated by regarding each vertex of the hexagon as the ideal center,the roundness error value of corresponding evaluation method (MZC,MCC and MIC) are obtained through comparison,judgment and repeated arrangement of hexagon.The experimental results show that the roundness error can be evaluated effectively and exactly by using this algorithm.

The relationship between the minimum zone circle and the maximum inscribed circle and the minimum circumscribed circle

Following the minimum zone criterion set forth in the current ANSI and ISO standards, evaluation of roundness error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The maximum inscribed circle and minimum circumscribed circle are all easily solved by iterative comparisons, so the relationship between the minimum zone circle and maximum inscribed circle, minimum circumscribed circle is proposed to solve efficiently the minimum zone problem. Based on the known minimum zone circle, the maximum inscribed circle and minimum circumscribed circle can be easily determined. The relationship is implemented and validated with the data available in the literature.

Applying particle swarm optimization algorithm to roundness measurement

Measuring the roundness of a circular workpiece is a common problem of quality control and inspection. In this area, maximum inscribed circle (MIC) and maximum circumscribing circle (MCC), minimum zone circle (MZC) and least square circle (LSC) are four commonly used methods. In particular, MIC, MCC, and MZC, which are nonlinear constrained optimization problems, have not been thoroughly discussed lately. This study proposes a machine vision-based roundness measuring method that applies the particle swarm optimization algorithm (PSO) to compute MIC, MCC and MZC. To facilitate the PSO process, five different PSO’s were encoded using a radius ( R) and circle center ( x, y) and extensively evaluated using an experimental design, in which the impact of inertia weight, maximum velocity and the number of particles on the performance of the particle swarm optimizer was analyzed. The proposed method was verified with a set of testing images and benchmarked with the GA-based (genetic algorithm) method [Chen, M. C. (2000). Roundness inspection strategies for machine visions using non-linear programs and genetic algorithms. International Journal of Production Research, 38, 2967–2988]. The experimental results reveal that the PSO-based method effectively solved the MIC, MCC, and MZC problems and outperforms GA-based method in both accuracy and the efficiency. As a finals, several industrial applications are presented to explore the effectiveness and efficiency of the proposed method.

Application of convex hull in the assessment of roundness error

The concept of the Minimum Zone Lines is proposed as a determinant condition of the minimum zone circle. The minimum zone roundness error from coordinate data is obtained by the distance between the Minimum Zone Lines. The Minimum Zone Lines determined by four critical measured points can be constructed based on convex hull and coordinates transmission. The rapid selection of iteration points with convex hull leads to an efficient way to solve the minimax solution. The objective of this strategy is to pick out the critical data points as early as possible so that the redundant data points can be rejected. The method developed is implemented and validated with the data available in the literature.

An effective genetic algorithm for circularity error unified evaluation

There are four methods commonly used to evaluate the circularity error. They are: minimum zone circle (MZC) method, minimum circumscribed circle (MCC) method, maximum inscribed circle (MIC) method and least square circle (LSC) method. However, so far there is no a robust and effective approach to implement a unified evaluation of these four methods. In this paper, an effective genetic algorithm is presented for searching the above four circularity error evaluation methods simultaneously. The algorithm is implemented in real-code and only blend crossover operators are applied to two randomly selected individuals from the existing population. The algorithm does not require genetic parameters such as crossover and mutation probabilities to be set in advance as does a canonical GA; therefore it is very convenient to use in engineering metrology. The objective function calculation approaches of four circularity errors are developed and the initial population generation methods are given in order to save optimization time. Finally, the experimental results evaluated by different methods confirm that the proposed method can find the optimal solutions of these four methods. In comparison with existing evaluation methods, the algorithm is not only simple and robust, but also it unifies these four kinds of circularity evaluation. The algorithm can also be used for solving difficult form error minimization and profile evaluation problems of various geometric parts in engineering metrology.

The min–max problem for evaluating the form error of a circle

There have been many studies to evaluate the form error of a circle. Most of them, such as the optimum methods and limacon model, employed the approximate solution to obtain the desired results. In this paper, three mathematical models depending on the method used to select the exact control points are constructed to evaluate the analytic solution of the minimum circumscribed circle, the maximum inscribed circle and the minimum zone circle by directly resolving the simultaneous linear algebraic equations. These new and simple mathematical methods are verified to be useful for determining the exact solution.

A stochastic optimization approach for roundness measurements

In this paper, we develop a vision-based inspection system for roundness measurements. A stochastic optimization approach has been proposed to compute the reference circles of MIC (maximum inscribing circle), MCC (minimum circumscribing circle) and MZC (minimum zone circle) methods. The proposed algorithm is a hybrid optimization method based on simulated annealing and Hooke–Jeeves pattern search. From the experimental results, it is noted that the algorithm can solve the roundness assessment problems effectively and efficiently. The developed vision-based inspection system can be an on-line tool for the measurement of circular components.

Application of neutral network interval regression method for minimum zone circle roundness

The goal of this work is to develop an accurate, efficient and robust algorithm for minimum zone circle roundness. In this paper, we use an interval bias adaptive linear neural network (NN) structure together with a least mean squares (LMS) learning algorithm, and an appropriate cost function to carry out the interval regression analysis. Through the system transformation, the minimum zone roundness can be related to a linear programming (LP) problem, which can be solved by the interval regression method. The interval regression method by NNs developed in this paper is applicable in linear regression analysis which has a complicated constraint, and where the least squares (LSQ) method cannot be used.

A remark on a characterization of out-of-roundness measures

Using a limacon approximation to linearize the minimum zone circle (MZC) problem, and then by applying duality theory of linear programming, Chetwynd (1980, 1985) obtained a geometric algorithm exhibiting an 'alternation property.' It is demonstrated that the alternation property can be obtained using a classical result due to Bonnesen (1924). The equivalence is demonstrated using elementary arguments.