Form Error Evaluation: An Iterative Reweighted Least Squares Algorithm*

2004 ◽  
Vol 126 (3) ◽  
pp. 535-541 ◽  
Author(s):  
Xiangyang Zhu ◽  
Han Ding ◽  
Michael Y. Wang
Keyword(s):  
Least Squares ◽  
Weighted Least Squares ◽  
Chebyshev Approximation ◽  
Approximation Problem ◽  
General Purpose ◽  
Geometric Features ◽  
Error Evaluation ◽  
Form Error ◽  
Minimum Zone ◽  
Effectiveness And Efficiency

This paper establishes the equivalence between the solution to a linear Chebyshev approximation problem and that of a weighted least squares (WLS) problem with the weighting parameters being appropriately defined. On this basis, we present an algorithm for form error evaluation of geometric features. The algorithm is implemented as an iterative procedure. At each iteration, a WLS problem is solved and the weighting parameters are updated. The proposed algorithm is of general-purpose, it can be used to evaluate the exact minimum zone error of various geometric features including flatness, circularity, sphericity, cylindericity and spatial straightness. Numerical examples are presented to show the effectiveness and efficiency of the algorithm.

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

2005 ◽  
Vol 128 (1) ◽  
pp. 167-174 ◽  
Author(s):  
LiMin Zhu ◽  
Ye Ding ◽  
Han Ding
Keyword(s):  
Linear Programming ◽  
Chebyshev Approximation ◽  
Approximation Problem ◽  
Linear Complex ◽  
Infinite Linear Programming ◽  
Minimum Zone ◽  
Effectiveness And Efficiency ◽  
Primal And Dual ◽  
Straightness Error ◽  
Infinite Linear

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.

scholarly journals Least-squares gradient calculation from multi-point observations of scalar and vector fields: methodology and applications with Cluster in the plasmasphere

2007 ◽  
Vol 25 (4) ◽  
pp. 971-987 ◽  
Author(s):  
J. De Keyser ◽  
F. Darrouzet ◽  
M. W. Dunlop ◽  
P. M. E. Décréau
Keyword(s):  
Vector Field ◽  
Least Squares ◽  
Measurement Errors ◽  
Vector Fields ◽  
Total Error ◽  
Weighted Least Squares ◽  
General Purpose ◽  
Small Scale ◽  
Divergence Free Vector ◽  
Gradient Calculation

Abstract. This paper describes a general-purpose algorithm for computing the gradients in space and time of a scalar field, a vector field, or a divergence-free vector field, from in situ measurements by one or more spacecraft. The algorithm provides total error estimates on the computed gradient, including the effects of measurement errors, the errors due to a lack of spatio-temporal homogeneity, and errors due to small-scale fluctuations. It also has the ability to diagnose the conditioning of the problem. Optimal use is made of the data, in terms of exploiting the maximum amount of information relative to the uncertainty on the data, by solving the problem in a weighted least-squares sense. The method is illustrated using Cluster magnetic field and electron density data to compute various gradients during a traversal of the inner magnetosphere. In particular, Cluster is shown to cross azimuthal density structure, and the existence of field-aligned currents in the plasmasphere is demonstrated.

Form Error Evaluation of Noncontact Scan Data Using Constriction Factor Particle Swarm Optimization

2017 ◽  
Vol 16 (03) ◽  
pp. 205-226 ◽  
Author(s):  
Vimal Kumar Pathak ◽  
Amit Kumar Singh
Keyword(s):  
Particle Swarm Optimization ◽  
Optimization Technique ◽  
Particle Swarm ◽  
Error Evaluation ◽  
Form Error ◽  
Swarm Optimization ◽  
Significant Difference ◽  
Minimum Zone ◽  
Measuring Machine ◽  
Constriction Factor

Form error evaluation of manufactured parts is one of the crucial aspects of precision coordinate metrology. With the advent of technology, the noncontact data acquisition techniques are replacing the conventional machines like coordinate measuring machine (CMM). This paper presents an optimization technique to evaluate minimum zone form errors, namely straightness, circularity, flatness and cylindricity using constriction factor-based particle swarm optimization (CFPSO) algorithm. Addition of constriction factor helps in accelerating the convergence property of CFPSO. Initially, a simple minimum zone objective function is formulated mathematically for each form error and then optimized using the proposed CFPSO. Primarily, the results of the proposed method for form error evaluation are compared with the literature results. Furthermore, the data obtained from noncontact 3D scanner is processed and the results of form error evaluation using CFPSO algorithm are compared with Steinbichler’s INSPECT PLUS software results. It was found that the results obtained using the proposed CFPSO algorithm are fast and better as compared with other evolutionary techniques like genetic algorithm (GA), previous literatures and software results. Furthermore, to ensure effectiveness of the proposed method statistical analysis ([Formula: see text]-test) was performed. CFPSO results for large dimension of problem show significant difference in computation time as compared with GA. The CFPSO algorithm provides 27.25%, 7.5% and 6.38% improvements in circularity, flatness and cylindricity, respectively, in comparison to RE software results, for determination of minimum zone error. Thus, the methodology presented helps in improving the accuracy and for speeding up of the automated inspection process generally performed by CMMs in industries.

Realization Method for Detection on Arc Based on CCD

2014 ◽  
Vol 687-691 ◽  
pp. 856-860
Author(s):  
Qing Min Liu ◽  
Xue Li ◽  
L. Zhang
Keyword(s):  
Least Squares ◽  
Evaluation Methods ◽  
Detection Methods ◽  
Inspection System ◽  
Distributed Data ◽  
Error Evaluation ◽  
Roundness Error ◽  
Vision Inspection ◽  
Minimum Zone ◽  
Least Squares Algorithm

s: Arc detection is difficult for processing, assembly and testing of industrial production because of limitations of detection methods, algorithms and instruments. The least-squares algorithm is used to fit data in circle detection. The application of conventional least-squares algorithm is limited, as roundness error is bigger, precision is lower. For detecting arc with data points of non-uniform distribution, improved least-squares algorithm, developed an analysis algorithm for assessing the minimum zone roundness error. Center and radius can be solved, without iteration and truncation error. Using the discrete data instances verified different roundness error evaluation methods. Visual measurements have been carried out using the proposed methods. Calculated results using the four kinds of roundness error evaluation methods (Figure 7-10). Ball diameter errors are-0.0245mm、0.0176mm、-0.1052mm and 0.302mm, roundness errors are 0.07mm、0.063mm、0.078mm and 0.146mm. The improved least-squares algorithm and the minimum zone algorithm are suitable for distributed data of all kinds situations, particularly suitable for the realization of machine vision inspection system, fast speed, high precision, wide application.

A Special Solution of Spatial Straightness for Four Datapoints

2012 ◽  
Vol 562-564 ◽  
pp. 544-547
Author(s):  
Fan Wu Meng ◽  
Chun Guang Xu ◽  
Juan Hao
Keyword(s):  
Linear System ◽  
Initial Conditions ◽  
Three Dimensional ◽  
The Novel ◽  
Form Error ◽  
Novel Approach ◽  
Minimum Zone ◽  
Effectiveness And Efficiency ◽  
Non Linear System ◽  
Straightness Error

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.

Parabola Error Evaluation Using Geometry Ergodic Searching Algorithm

2013 ◽  
Vol 333-335 ◽  
pp. 1465-1468
Author(s):  
Hai Yang Wang ◽  
Xian Qing Lei ◽  
Jing Wei Cui
Keyword(s):  
Least Square Method ◽  
Least Square ◽  
Geometric Features ◽  
Feature Points ◽  
Error Evaluation ◽  
Profile Error ◽  
Minimum Zone ◽  
Grid Points ◽  
Reference Feature

A method of parabola error evaluation using Geometry Ergodic Searching Algorithm (GESA) was proposed according to geometric features and fitting characteristics of parabola error. First , the feature points of least-squared parabola are set as reference feature points to layout a group of auxiliary feature grid points. After that, a series of auxiliary parabolas as assumed ideal parabolas are reversed with the auxiliary feature points.The range distance from given points to these assumptions ideal parabolas are calculated successively.The minimum one is parabola profile error.The process of GESA was detailed discribed including the algorithm formula and contrastive results in this paper.Simulation experiment results show that the geometry ergodic searching algorithm is more accurate than the least-square method. The parabola profile error can be evaluated steadily and precisely with this algorithm based on the minimum zone.

Evaluation for estimating of the PDF and the CDF of Generalized Inverted Exponential Distribution with Application in Industry

2020 ◽  
pp. 507-522
Author(s):  
Parisa Torkaman
Keyword(s):  
Least Squares ◽  
Exponential Distribution ◽  
Mean Squared Error ◽  
Weighted Least Squares ◽  
Real Data ◽  
Minimum Variance ◽  
Cumulative Distribution ◽  
Estimation Methods ◽  
Data Set ◽  
Better Than

The generalized inverted exponential distribution is introduced as a lifetime model with good statistical properties. This paper, the estimation of the probability density function and the cumulative distribution function of with five different estimation methods: uniformly minimum variance unbiased(UMVU), maximum likelihood(ML), least squares(LS), weighted least squares (WLS) and percentile(PC) estimators are considered. The performance of these estimation procedures, based on the mean squared error (MSE) by numerical simulations are compared. Simulation studies express that the UMVU estimator performs better than others and when the sample size is large enough the ML and UMVU estimators are almost equivalent and efficient than LS, WLS and PC. Finally, the result using a real data set are analyzed.

The minimum zone fitting and error evaluation for the logarithmic curve based on geometry optimization approximation algorithm

2019 ◽  
Vol 41 (15) ◽  
pp. 4380-4386
Author(s):  
Tu Xianping ◽  
Lei Xianqing ◽  
Ma Wensuo ◽  
Wang Xiaoyi ◽  
Hu Luqing ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Evaluation Method ◽  
Geometry Optimization ◽  
Reference Points ◽  
Approximation Technique ◽  
Error Evaluation ◽  
Iterative Approximation ◽  
Logarithmic Curve ◽  
Normal Distance ◽  
Minimum Zone

The minimum zone fitting and error evaluation for the logarithmic curve has important applications. Based on geometry optimization approximation algorithm whilst considering geometric characteristics of logarithmic curves, a new fitting and error evaluation method for the logarithmic curve is presented. To this end, two feature points, to serve as reference, are chosen either from those located on the least squares logarithmic curve or from amongst measurement points. Four auxiliary points surrounding each of the two reference points are then arranged to resemble vertices of a square. Subsequently, based on these auxiliary points, a series of auxiliary logarithmic curves (16 curves) are constructed, and the normal distance and corresponding range of values between each measurement point and all auxiliary logarithmic curves are calculated. Finally, by means of an iterative approximation technique consisting of comparing, evaluating, and changing reference points; determining new auxiliary points; and constructing corresponding auxiliary logarithmic curves, minimum zone fitting and evaluation of logarithmic curve profile errors are implemented. The example results show that the logarithmic curve can be fitted, and its profile error can be evaluated effectively and precisely using the presented method.

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