Minimum Zone Criterion Research Articles

Method of inclined hole axis adjustment based on geometric optimization and minimum zone theory

The machining accuracy of the axis position and spatial angle of the array microholes with high aspect ratio on the components of the high-end equipment manufacturing industry will directly affect the performance of the whole machine. A method of tilt hole axis adjustment is proposed in this paper. It utilizes geometric optimization and minimum zone theory to measure the position and spatial angle of the hole axis. This addresses the slight angle error that occurs after correcting the inclined hole axis using visual techniques. Based on the minimum zone criterion for the cylindricity error, the evaluation model is re-established through the inner and the outer loop and spatial projection, which includes roundness evaluation of the projection curve by an improved particle swarm optimization (PSO) algorithm and elliptic fitting by a geometric parameter method, so as to obtain the optimal solution of the axis vector and spatial angle. Experimental comparison with the theory above shows that the precision of the method proposed is better than that of the least squares method and the traditional PSO algorithm based on the minimum zone criterion. The research results indicate that the precision of the tilt angle is better than 0.03°, and simultaneously the rotation angle is better than 0.1° in the range of 1°. This experiment verifies the effectiveness and accuracy of the proposed method.

An Adaptive Multi-Population Approach for Sphericity Error Evaluation in the Manufacture of Hemispherical Shell Resonators.

The performance of a hemispherical resonant gyroscope (HRG) is directly affected by the sphericity error of the thin-walled spherical shell of the hemispherical shell resonator (HSR). In the production process of the HSRs, high-speed, high-accuracy, and high-robustness requirements are necessary for evaluating sphericity errors. We designed a sphericity error evaluation method based on the minimum zone criterion with an adaptive number of subpopulations. The method utilizes the global optimal solution and the subpopulations' optimal solution to guide the search, initializes the subpopulations through clustering, and dynamically eliminates inferior subpopulations. Simulation experiments demonstrate that the algorithm exhibits excellent evaluation accuracy when processing simulation datasets with different sphericity errors, radii, and numbers of sampling points. The uncertainty of the results reached the order of 10-9 mm. When processing up to 6000 simulation datasets, the algorithm's solution deviation from the ideal sphericity error remained around -3 × 10-9 mm. And the sphericity error evaluation was completed within 1 s on average. Additionally, comparison experiments further confirmed the evaluation accuracy of the algorithm. In the HSR sample measurement experiments, our algorithm improved the sphericity error assessment accuracy of the HSR's inner and outer contour sampling datasets by 17% and 4%, compared with the results given by the coordinate measuring machine. The experiment results demonstrated that the algorithm meets the requirements of sphericity error assessment in the manufacturing process of the HSRs and has the potential to be widely used in the future.

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.

A Registration Method for Profile Error Inspection of Complex Surface Under Minimum Zone Criterion

Although registration of the measured model and the design model has been widely studied, it remains an open problem in complex surface profile error inspection. False rejection of some qualified parts is prone to occur due to the improper registration between the measured points and the nominal model. To solve this problem, we define a surface profile error metric in terms of a new range norm under minimum zone criterion. An optimal framework based on the range norm is proposed to find the registration pose of the measurement data set, in which the surface profile error is minimized. To deal with the computational intractability, the non-differentiable merit function is uniformly approximated by a smooth aggregation function, which can be effectively solved with the limited-memory Quasi-Newton method. Distinct numerical stability control and the active set selection mechanism are proposed to guarantee the accuracy and efficiency of the method. Experiments on a simulated and a real part are included to verify the superiority of the proposed method.

Profile error evaluation of free-form surface using sequential quadratic programming algorithm

Profile error of free-form surface is evaluated in this paper based on sequential quadratic programming (SQP) algorithm. The optimal localization model is established with the minimum zone criterion firstly. Subsequently, the surface subdivision method or STL (STeror Lithography) model is used to compute the point-to-surface distance and the approximate linear differential movement model of signed distance is deduced to simplify the updating process of alignment parameters. Finally, the optimization model on profile error evaluation of free-form surface is solved with SQP algorithm. Simulation examples indicate that the results acquired by SQP method are closer to the ideal results than the other algorithms in the problem of solving transformation parameters. In addition, real part experiments show that the maximum distance between the measurement points and their corresponding closest points on the design model is shorter by using SQP-based algorithm. Lastly, the results obtained in the experiment of the workpiece with S form illustrate that the SQP-based profile error evaluation algorithm can dramatically reduce the iterations and keep the precision of result simultaneously. Furthermore, a simulation is conducted to test the robustness of the proposed method. In a word, this study purposes a new algorithm which is of high accuracy and less time-consuming.

Comparison of GUF and Monte Carlo methods to evaluate task-specific uncertainty in laser tracker measurement

Measurement uncertainty plays an important role in laser tracking measurement analyses. In the present work, the guides to the expression of uncertainty in measurement (GUM) uncertainty framework (GUF) and its supplement, the Monte Carlo method, were used to estimate the uncertainty of task-specific laser tracker measurements. First, the sources of error in laser tracker measurement were analyzed in detail, including instruments, measuring network fusion, measurement strategies, measurement process factors (such as the operator), measurement environment, and task-specific data processing. Second, the GUM and Monte Carlo methods and their application to laser tracker measurement were presented. Finally, a case study involving the uncertainty estimation of a cylindricity measurement process using the GUF and Monte Carlo methods was illustrated. The expanded uncertainty results (at 95% confidence levels) obtained with the Monte Carlo method are 0.069 mm (least-squares criterion) and 0.062 mm (minimum zone criterion), respectively, while with the GUM uncertainty framework, none but the result of least-squares criterion can be got, which is 0.071 mm. Thus, the GUM uncertainty framework slightly underestimates the overall uncertainty by 10%. The results demonstrate that the two methods have different characteristics in task-specific uncertainty evaluations of laser tracker measurements. The results indicate that the Monte Carlo method is a practical tool for applying the principle of propagation of distributions and does not depend on the assumptions and limitations required by the law of propagation of uncertainties (GUF). These features of the Monte Carlo method reduce the risk of an unreliable measurement of uncertainty estimation, particularly in cases of complicated measurement models, without the need to evaluate partial derivatives. In addition, the impact of sampling strategy and evaluation method on the uncertainty of the measurement results can also be taken into account with Monte Carlo method, which plays a guiding role in measurement planning.

1G3 最小領域基準に基づく回転曲面輪郭の形状偏差評価([GS]設計・トライボロジーI)

1G3 最小領域基準に基づく回転曲面輪郭の形状偏差評価([GS]設計・トライボロジーI)

Conicity error evaluation using sequential quadratic programming algorithm

• Least-square method is employed to locate the cone roughly. • Simplified signed distance function is derived. • Simplified differential motion model of distance function is deduced. • SQP evaluation algorithm of conicity error is obtained. • The evaluation algorithm applies to cone with arbitrary position and orientation. Form error evaluation plays an important role in processing quality evaluation. Conicity error is evaluated as a typical example in this paper based on sequential quadratic programming (SQP) algorithm. The evaluation is carried out in three stages. Signed distance function from the measured points to conical surface is defined and the cone is located roughly by the method of traditional least-squares (LS) firstly; the fitted cone and the measured point coordinates are transformed to simplify the optimal mathematical model of conicity error evaluation secondly; and then optimization problem on conicity error evaluation satisfying the minimum zone criterion is solved by means of SQP algorithm and kinematic geometry, where approximate linear differential movement model of signed distance function is deduced in order to reduce the computational complexity. Experimental results show that the conicity error evaluation algorithm is more accurate, and has good robustness and high efficiency. The obtained conicity error is effective.

Cutter size optimisation and interference-free tool path generation for five-axis flank milling of centrifugal impellers

Strategies for cutter size optimisation and interference-free tool path generation are proposed for five-axis flank milling of centrifugal impellers. To increase the material removal rate and provide a stronger tool shank during flank milling, the cutter size is first maximised under a set of geometric constraints. The tool path is then globally optimised in accordance with the minimum zone criterion for the determined optimal cutter size. Aside from the local interference of the cutter with the design surface, the global interferences with the hub surface and the adjacent blade surface are also considered in the optimisation models. Interference is indicated by the signed distance from the sampled point on the blade surface to the tool envelope surface. This distance is calculated without constructing the envelope surface. On the basis of the differential property of the distance function, we choose a sequential linear programming method in implementing the optimisations. This approach applies to generic rotary cutters, such as cylindrical and conical tools. Simulations are conducted to obtain the optimal cutter size and generate an interference-free tool path for a practical impeller. Simultaneously, a software module that can generate tool envelope surfaces and verify geometric errors is used to validate the proposed method.

Geometry of Signed Point-to-Surface Distance Function and Its Application to Surface Approximation

This paper presents a unified framework for computing a surface to approximate a target shape defined by discrete data points. A signed point-to-surface distance function is defined, and its properties are investigated, especially, its second-order Taylor approximant is derived. The intercorrelations between the signed and the squared distance functions are clarified, and it is demonstrated that the squared distance function studied in the previous works is just the Type I squared distance function deduced from the signed distance function. It is also shown that surface approximations under different criteria and constraints can all be formulated as optimization problems with specified requirements on the residual errors represented by the signed distance functions, and that classical numerical optimization algorithms can be directly applied to solve them since the derivatives of the involved objective functions and constraint functions are all available. Examples of global cutter position optimization for flank milling of ruled surface with a cylindrical tool, which requires approximating the tool envelope surface to the point cloud on the design surface following the minimum zone criterion, are given to confirm the validity of the proposed approach.

Global optimization of tool path for five-axis flank milling with a conical cutter

In this paper, optimum positioning of the conical cutter for five-axis flank milling of slender surfaces is addressed from the perspective of approximating the tool envelope surface to the data points on the design surface following the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation. Based on the observation that a conical surface can be treated as a canal surface, i.e. envelope surface of one-parameter family of spheres, the swept envelope of a conical cutter is represented as a sphere-swept surface. Then, an approach is presented to efficiently compute the signed distance between a point in space and the swept surface without constructing the swept surface itself. The first order differential increment of the signed point-to-surface distance with respect to the differential deformation of the tool axis trajectory surface is derived. By using the distance function, tool path optimizations for semi-finish and finish millings are formulated as two constrained optimization problems in a unified framework, and a sequential approximation algorithm along with a hierarchical algorithmic structure is developed for the optimization. Numerical examples are given to confirm the validity and efficiency of the proposed approach. Comparing with the existing approaches, the present one improves the machining accuracy greatly. The rationale developed applies to general rotary cutters.

Measurement and evaluation of form deviation error of disk cam with an exclusively built profile-measuring machine

To evaluate the manufacturing precision of disk cams, an effective measurement method with an exclusively built profile-measuring machine and subsequent data analysis procedure is proposed. The profile-measuring machine consisted of a rotating table and sensing devices interfaced to a PC. This machine could provide sufficient measured data with just one revolving scan of the profile for evaluation of the form deviation error of disk cams. The design and measurement data were interpolated by cubic spline curves to compute the manufacturing precision error, which is defined by the maximum and minimum distances between two curves. The minimum zone criterion (MZC) of ISO was employed to evaluate the form deviation error. The proposed system was applied to marine engine cams, and it required only 5 min to measure and analyze the form deviation error of a cam. With a conventional coordinate measuring machine (CMM), the same error measurement would require over 1 h even with a skilled operator.

Minimum zone evaluation of conicity error using minimum potential energy algorithms

Current algorithm system of the Coordinate Measuring Machine (CMM) widely adopts the least-squares (LS) method for form error quality inspections. This LS algorithm provides an approximate solution for form error evaluation only, with possible actual error overestimation and further leading some of acceptable mechanical components to be rejected. Moreover, the existing method for the evaluation of conicity error is actually infrequent. Thus, the paper elaborates a heuristic approach using an umbrella-shaped mechanical model for precisely evaluating the minimum zone conicity error based on minimum potential energy principle. The umbrella-shaped mechanical model consists of two coaxial and equal-vertex-angle conical surfaces by a fictitious spring connection. All measured data points are enclosed within two conical surfaces which are mathematically determined by the coordinates of 7 active data points. To allow the error assessment being conducted precisely, the non-linear model provides seven degrees of freedom to handle various possible situations. The shrinking spring reduces simulated mechanical system potential energy, yielding two new coaxial cones with smaller normal separation and new contact points on conical surfaces. The system eventually reaches a stable state with a minimum elastic potential energy. The normal separation between such two conical surfaces is minimum zone of conical form error. A direct searching technique based on the derived minimum zone criterion is demonstrated, and a fast and flexible computational algorithm is also proposed.

Global optimization of tool path for five-axis flank milling with a cylindrical cutter

In this paper, optimum positioning of cylindrical cutter for five-axis flank milling of non-developable ruled surface is addressed from the perspective of surface approximation. Based on the developed. interchangeability principle, global optimization of the five-axis tool path is modeled as approximation of the tool envelope surface to the data points on the design surface following the minimum zone criterion recommended by ANSI and ISO standards for tolerance evaluation. By using the signed point-to-surface distance function, tool path plannings for semi-finish and finish millings are formulated as two constrained optimization problems in a unified framework. Based on the second order Taylor approximation of the distance function, a sequential approximation algorithm along with a hierarchical algorithmic structure is developed for the optimization. Numerical examples are presented to confirm the validity of the proposed approach.

Tool path optimisation for flank milling ruled surface based on the distance function

This paper deals with optimised tool path generation for five-axis flank milling using signed point-to-surface distance function. The main idea is that the geometrical deviations between the design surface and the machined surface are minimised by fine tuning the cutter locations. Based on the tangency conditions in envelope theory, the analytic representation of the envelope surface of a cutter undergoing five-axis motion is first obtained. Then the geometrical deviations between the envelope surface (i.e. machined surface) and the design surface are calculated. Optimisation of the five-axis tool path is modeled as the fine tuning of the initial cutter locations under the minimum zone criterion recommended by ANSI and ISO, which requires minimisation of the maximum geometrical deviation between the design surface and the envelope surface. Using the signed point-to-surface distance function, tool path optimisation for finish milling is formulated as a constrained optimisation problem. Based on the first-order Taylor approximation of the signed distance function, two sequential approximation algorithms for the Minimax and Least Square optimisations are developed. Numerical examples, in which a conical tool is chosen as a special case of flank machining ruled surface, verify the proposed strategy.

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.

A unified approach for circularity and spatial straightness evaluation using semi-definite programming

This paper demonstrates that a modern optimization method known as semi-definite programming (SDP) can serve as a unified tool for circularity and spatial straightness evaluation. Based on the minimum zone criterion (MZC), the problems of circularity and spatial straightness evaluation are formulated as differentiable constrained optimization problems firstly, and then reformulated as SDP problems. Within the SDP framework, both of the two aforementioned problems can be solved efficiently using recently developed interior-point algorithms, which are available in many efficient software packages. The effectiveness and efficiency of the novel approach are illustrated by examples and numerical simulations.

Algorithm for Spatial Straightness Evaluation Using Theories of Linear Complex Chebyshev Approximation and Semi-infinite Linear Programming

Abstract This paper presents a novel methodology for evaluating spatial straightness error based on the minimum zone criterion. Spatial straightness evaluation is formulated as a linear complex Chebyshev approximation problem, and then reformulated as a semi-infinite linear programming problem. Both models for the primal and dual programs are developed. An efficient simplex-based algorithm is employed to solve the dual linear program to yield the straightness value. Also a general algebraic criterion for checking the optimality of the solution is proposed. Numerical experiments are given to verify the effectiveness and efficiency of the presented algorithm.

Evaluation of angular error between two lines

The error caused by the fitting of a set of data points to two lines having a specified angle is defined as the angular error. The evaluation of angular error needs to find two pairs of parallel lines that follow the angular constraint and bound the data points under the minimum zone criterion. The evaluation of angular error is difficult in mathematics, so it is usually simplified by treating one line as the datum and the form error of the other line calculated from the datum as the solution. The datum is assumed to be perfect that doesn’t tell the real properties of the actual line but gives incorrect solutions in general. To give an exact solution to the angular error problem based on the minimum zone criterion, a new mathematical model is proposed in this paper. The basic idea of this model is to rotate the data points of one line to the same direction as that of the other line so that the evaluation of angular error is simplified as solving a straightness problem. The angular error obtained from the proposed model is proved to be minimum. The proposed model offers a simple approach to solve the tough angular error problem, and it also provides a feasible tool to explore more complicated problems, like 3D angular error and polygon error.

A steepest descent algorithm for circularity evaluation

This paper presents a novel algorithm for evaluating the circularity of a mechanical part by using measurement points obtained with a coordinate measuring machine (CMM). Following the minimum zone criterion set forth in the current ANSI and ISO standards, evaluation of circularity is formulated as a non-differentiable unconstrained optimization problem, and based on the geometric representation of the necessary and sufficient condition for the optimal solution, an efficient steepest descent optimization procedure is proposed to find the circularity value. The steepest descent direction is determined by the method of calculating the minimum translational distance between two convex polygons, which is initially introduced in the field of robot path planning, and the length of the moving step is exactly determined by a presented geometrical method. A computational geometry-based method for pre-processing the measured data is also proposed. In comparison with existing methods, this algorithm has the advantages of computational efficiency and high precision. Simulations and practical example confirm the validity of the presented algorithm.