Spatial Straightness Error Research Articles

Study on straightness error evaluation of spatial lines based on a hybrid ant colony algorithm

Based on the analysis of existing evaluation methods for straightness errors, an intelligent evaluation method for spatial straightness errors is provided in this paper. The evolutional optimum model and the calculation process are introduced in detail. According to characteristics of straightness error for spatial lines evaluation, an improved Ant Colony Optimisation ACO algorithm is proposed to evaluate the minimum zone error. Compared with conventional optimum evaluation methods such as Simplex search and Powell method, it can find the global optimal solution and the precision of calculating result is very high. Then, the objective function calculation approaches for using the ACO to evaluate minimum zone error are formulated. Finally, the control experiment results evaluated by different optimal methods such as the Least Square, Simplex search, Powell search and Genetic Algorithm, indicate that the proposed method does provide better accuracy on spatial straightness error evaluation and it has fast convergent speed as well as using computer.

Spatial straightness error evaluation based on three-dimensional least squares method

Spatial straightness error evaluation based on three-dimensional least squares method

Straightness error compensation for large CNC gantry type milling centers based on B-spline curves modeling

Fast and accurate modeling for the straightness errors of machine tools is significant important for the implementation of error compensation. To decrease the straightness errors which generally exist in gantry type milling centers, a novel approach for real-time compensation of straightness errors is presented, including an accurate spatial straightness error model which based on B-Spline curves method and a real-time errors compensation system which can compensate multiple errors at the same time in real-time. An experiment is carried out to measure the straightness errors of a gantry type milling center by using a laser interferometer. There are two straightness errors for one direction movement of an axis. As a result, there are six straightness errors for a three-axis milling center. The straightness error model is established by the B-Spline curves method. According to these six straightness error models, a spatial straightness error model is obtained by superposition of these straightness error models. In addition, an external real-time errors compensation system is developed based on the function of external mechanical origin offset in Fanuc CNC systems. The compensation experiments are conducted to verify the accuracy of the spatial straightness error model and the effectiveness of the error compensation system. The experimental results show that the straightness errors of the machined work-pieces are proved to have been decreased over 90% compared with that of non-compensations. The B-Spline curves modeling combining the errors compensation system can be utilized as an effective approach to improve the accuracy of the CNC gantry type milling centers.

Geometric Approximation Searching Algorithm for Spatial Straightness Error Evaluation

Considering the characteristics of spatial straightness error, this paper puts forward a kind of evaluation method of spatial straightness error using Geometric Approximation Searching Algorithm (GASA). According to the minimum condition principle of form error evaluation, the mathematic model and optimization objective of the GASA are given. The algorithm avoids the optimization and linearization, and can be fulfilled in three steps. First construct two parallel quadrates based on the preset two reference points of the spatial line respectively; second construct centerlines by connecting one quadrate each vertices to another quadrate each vertices; after that, calculate the distances between measured points and the constructed centerlines. The minimum zone straightness error is obtained by repeating comparing and reconstructing quadrates. The principle and steps of the algorithm to evaluate spatial straightness error is described in detail, and the mathematical formula and program flowchart are given also. Results show that this algorithm can evaluate spatial straightness error more effectively and exactly.

Research on a Computerized Form and Position Error Measurement Instrument

A model machine of multifunctional form and position measurement instrument controlled by a personal computer has been successfully developed. The instrument is designed in rotary table type with a high precision air bearing and the radial rotation error of the rotary table is 0.08 μm. A high precision vertical sliding carriage supported by an air bearing is used to the instrument, the straight motion error of the carriage is 0.3 μm/200 mm and the parallelism error of the motion of the carriage relative to the rotation axis of the rotary table is 0.4 μm/200 mm. The mathematical models have been established for assessing planar and spatial straightness, flatness, roundness, cylindricity, and coaxality errors. By radial deviation measurement, the instrument can accurately measure form and position errors of such workpieces as shafts, round plates and sleeves of medium or small dimensions with the tolerance grades mostly used in industry.

Monte carlo method for the uncertainty evaluation of spatial straightness error based on new generation geometrical product specification

Straightness error is an important parameter in measuring high-precision shafts. New generation geometrical product specification(GPS) requires the measurement uncertainty characterizing the reliability of the results should be given together when the measurement result is given. Nowadays most researches on straightness focus on error calculation and only several research projects evaluate the measurement uncertainty based on “The Guide to the Expression of Uncertainty in Measurement(GUM)”. In order to compute spatial straightness error(SSE) accurately and rapidly and overcome the limitations of GUM, a quasi particle swarm optimization(QPSO) is proposed to solve the minimum zone SSE and Monte Carlo Method(MCM) is developed to estimate the measurement uncertainty. The mathematical model of minimum zone SSE is formulated. In QPSO quasi-random sequences are applied to the generation of the initial position and velocity of particles and their velocities are modified by the constriction factor approach. The flow of measurement uncertainty evaluation based on MCM is proposed, where the heart is repeatedly sampling from the probability density function(PDF) for every input quantity and evaluating the model in each case. The minimum zone SSE of a shaft measured on a Coordinate Measuring Machine(CMM) is calculated by QPSO and the measurement uncertainty is evaluated by MCM on the basis of analyzing the uncertainty contributors. The results show that the uncertainty directly influences the product judgment result. Therefore it is scientific and reasonable to consider the influence of the uncertainty in judging whether the parts are accepted or rejected, especially for those located in the uncertainty zone. The proposed method is especially suitable when the PDF of the measurand cannot adequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution and the measurement model is non-linear.

A Special Solution of Spatial Straightness for Four Datapoints

Assessment of spatial straightness error is one of the most difficult tasks in form error evaluation, especially when the minimum zone three-dimensional (3D) straightness is decided by four datapoints. This paper develops a solution that can transform the non-linear system into a linear system. To solve the linear indeterminate system of equations, a method of factor fixed is proposed. Given adequate initial conditions, this solution will be the exact solution. The effectiveness and efficiency of the novel approach are illustrated by two examples.

A Linear Algorithm of Spatial Straightness for Five Points

It is a non-linear problem to evaluate the minimum zone spatial (3-D) straightness. Because of the disadvantages of evaluating spatial straightness error with traditional method, such as difficulty in solving the non-linear equation, the evaluating precision is lower. This paper develops a solution that can transform the non-linear problem into a linear problem. The solution can obtain more exact results than traditional method using results of LSC as initial conditions. This methodology can also be applied to the problems of computing the minimum circumscribed cylinder and the maximum inscribed cylinder. The effectiveness and efficiency of the novel approach are illustrated by two examples.

A combinatorial optimization approach for evaluating minimum-zone spatial straightness errors

This paper presents a new and robust approach for the accurate evaluation of minimum-zone spatial straightness error from a set of coordinate measurement data points. The algorithm iteratively searches for the specific data points that define the minimum bound of the spatial straightness zone using combinatorial optimization. It is based on the fact that the minimum circumscribed cylinder of a point set, which is equivalent to the minimum spatial straightness zone of the measurement data, will pass through three, four, or five of the data points that constitute the convex hull vertices of the entire data set. Computed results have shown that although the presented approach may lead to increased computational time, it is robust and able to construct the exact minimum circumscribed cylinder for a given point set. The minimum-zone spatial straightness error can thus be evaluated with the best possible accuracy. The advantage of the presented algorithm is demonstrated via comparison with published computed results of existing algorithms.

New Method for Evaluating Arbitrary Spatial Straightness Error

Because of the disadvantages in evaluating spatial straightness error with traditional method,such as difficulty in solving the nonlinear simultaneous equation,the evaluating precision is lower and data processing can't be automated.New method for evaluating arbitrary spatial straightness error is put forward.The particular data processing method and evaluating method are put forward,and the feasibility of method is validated by practical measurement.The result indicates that the method is simple and easy for automatic data processing by computer,and the evaluating precision is enhanced by 5% compared to the traditional method.

Semidefinite programming for Chebyshev fitting of spatial straight line with applications to cutter location planning and tolerance evaluation

This paper presents a novel model for fitting of spatial straight line based on Chebyshev norm. The problem is firstly formulated as a minimax problem, and then reformulated as a semidefinite programming (SDP) problem, which could be solved by many interior-point algorithms. The application of the proposed approach to two problems arising from manufacturing engineering, i.e. planning of the initial location of cylindrical cutter for flank milling and evaluation of the spatial straightness error, is discussed. Examples and numerical simulations illustrate the efficiency of the novel model.

The Theory and Assessment of Spatial Straightness Error Matched New Generation GPS

In order to assess spatial straightness error matched new generation Dimensional Geometrical Product Specification and Verification (GPS), the theory of spatial straightness error assessing is proposed and its advantages are analyzed based on metrology and statistics in this paper. Then, the assessing parameter system is proposed and it is testified in real application comparing to assessment result of the geometric tolerance theory. Statistical parameters of this assessing system post the different characteristics of spatial straightness error, and can reveal the impact of spatial straightness error on the accessory function more roundly to complement the single assessing parameter of geometrical tolerance for straightness error. The statistical spatial straightness tolerance and statistical spatial straightness error proposed in this paper is possible to be applied in evaluation of other error of form, orientation, location and run-out.

Development of a Computerized Multifunctional Form and Position Measurement Instrument

A model machine of multifunctional form and position measurement instrument controlled by a personal computer has been successfully developed. The instrument is designed in rotary table type with a high precision air bearing and the radial rotation error of the rotary table is 0.08 μm. Since a high precision vertical sliding carriage supported by an air bearing is used for the instrument, the straightaway motion error of the carriage is 0.3 μm/200 mm and the parallelism error of the motion of the carriage relative to the rotation axis of the rotary table is 0.4 μm/200 mm. The mathematical models have been established for assessing planar and spatial straightness, flatness, roundness, cylindricity, and coaxality errors. By radial deviation measurement, the instrument can accurately measure form and position errors of such workpieces as shafts, round plates and sleeves of medium or small dimensions with the tolerance grades mostly used in industry.

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.

A straightness error measurement method matched new generation GPS

The axis of the non-diffracting beam produced by an axicon is very stable and can be adopted as the datum line to measure the spatial straightness error in continuous working distance, which may be short, medium or long. Though combining the non-diffracting beam datum-line with LVDT displace detector, a new straightness error measurement method is developed. Because the non-diffracting beam datum-line amends the straightness error gauged by LVDT, the straightness error is reliable and this method is matchs new generation GPS.

Measurement method of spatial straightness error using non-diffracting beam and moiré-fringe technology

Based on the datum-line of the non-diffracting beam produced by an axicon, which is a thin lens with a weak conical surface on one face, a new measurement method of spatial straightness error has been developed in this paper, in combination with moire-fringe technology. When a ring grating is illuminated by a non-diffracting beam, a moire-fringe image is formed, from which the deviation of the centre of the ring grating to the centre of the Bessel fringe ring on the section of the non-diffracting beam, the offset, can be computed. The centre of the Bessel fringe ring embodies the non-diffracting beam datum-line, so the offset is also to the non-diffracting beam datum-line. Then, the spatial straightness error can be obtained from the set of the offsets corresponding to the section along the non-diffracting beam. In order to verify the performance of the new method, theoretical and experimental analyses were conducted. The results show that the presented method is effective and possesses a high system resolution in measuring the spatial straightness error. The measurement uncertainty is close to 0.28 µm in the measuring distance range 0.4–1.8 m behind the axicon.

An improved genetic algorithm for planar and spatial straightness error evaluation

The measurement data obtained from the Coordinate Measuring Machines (CMMs) have to be further processed and analyzed to evaluate the form errors of manufactured components. An improved genetic algorithm (GA) is proposed to implement the minimum zone evaluation of planar and spatial straightness errors simultaneously. The algorithm employs the generation alternation model based Minimal Generation Gap (MGP) and blend crossover operators (BLX- α). Compared to traditional GAs, it is efficient and robust. Then, the objective function calculation approaches of planar and spatial straightness error are developed, which directly originate from the definition of minimum zone solution and conform to the ISO standard. Finally, the experimental results evaluated by different methods confirm the effectiveness of the proposed GA. Compared to conventional evaluation methods; it has the advantages of algorithm simplicity and good flexibility. Also it is a unified approach for other form error evaluations and is well suited for the form error evaluation on CMMs.

Distance function based algorithm for spatial straightness evaluation

This paper focuses on the development of an efficient algorithm for evaluating the spatial straightness error based on the minimum circumscribed cylinder (MCC). A signed point-surface distance function is defined, and its increment with respect to the differential motion of the surface is derived. On this basis, spatial straightness evaluation is formulated as a non-linear constrained optimization problem, and a sequential approximation algorithm, with the magnitude of the differential motion of the reference axis being well controlled at each iteration, is developed to solve it. Also, a simple approach for verification of the optimality of the solution is presented. compared with existing algorithms, it has the advantages of implementational simplicity, computational efficiency and robustness. Examples confirm the validity of the proposed algorithm.

Study of the tolerance of the non-diffracting beam to laser beam deflection

Because of the large beam diameter and special characteristics of the non-diffracting beam produced by an axicon, the non-diffracting beam can tolerate the eccentricity of the incident laser beam's axis to the axicon. The non-diffracting beam obtains the additional tolerance property to the incident light beam's deflection when adopting a kind of incident divergent sphere Gauss wave, and this is proved in this paper through theoretical analysis and practical tests. The obtained conclusion is that the axis of the non-diffracting beam produced by an axicon is very stable and can be adopted as the datum line to measure the spatial straightness error in continuous working distance, which may be short, medium or long.

Evaluation method for spatial straightness errors based on minimum zone condition

A nonlinear mathematical model for spatial straightness error evaluation based on the minimum zone condition is established in this paper. According to the error analysis, it is proved that the mathematical model for spatial straightness error evaluation cannot be linearized. A criterion for verification of the existence and uniqueness of the minimum zone solution is proposed. A new computational method is also proposed, and practical examples are given. Finally, the correctness of this method is demonstrated using a geometrical solution. This new method is convenient for computation of uniqueness and exactness of the minimum zone solution.