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Maximum permissible errors (MPEs) are an important measurement system specification and form the basis of periodic verification of a measurement system’s performance. However, there is no standard methodology for determining MPEs, so when they are not provided, or not suitable for the measurement procedure performed, it is unclear how to generate an appropriate value with which to verify the system. Whilst a simple approach might be to take many measurements of a calibrated artefact and then use the maximum observed error as the MPE, this method requires a large number of repeat measurements for high confidence in the calculated MPE. Here, we present a statistical method of MPE determination, capable of providing MPEs with high confidence and minimum data collection. The method is presented with 1000 synthetic experiments and is shown to determine an overestimated MPE within 10% of an analytically true value in 99.2% of experiments, while underestimating the MPE with respect to the analytically true value in 0.8% of experiments (overestimating the value, on average, by 1.24%). The method is then applied to a real test case (probing form error for a commercial fringe projection system), where the efficiently determined MPE is overestimated by 0.3% with respect to an MPE determined using an arbitrarily chosen large number of measurements.

Coordinate Measuring Machines(CMM) are widely used in form inspection of free-form surfaces. Generally, the form error at each measured point is estimated using the widely known and accurate point-inversion method. This method has relatively high time complexity and cannot be preferred for fast inspection. Hence in this work, an alternative two-stage methodology based on the concept of the Voronoi diagram is proposed. In the first stage, the poles data is extracted from the Voronoi diagram of the discretized surface. In the second stage, the form-error-estimate algorithm executing in O(mlogn) time estimates the errors using the poles data and the discretized surface. Numerical and experimental implementations are executed using NURBS surfaces. The proposed method’s accuracy is on par with the point-inversion method and is 94.97% faster than the latter. Hence this method can be used for fast and accurate CMM and CNC based(in-situ) free-form surface inspection.

This paper is concerned with the generation of numerical artefacts, that is, reference data sets, for assessing the accuracy and fitness of purpose of software for computing minimum zone (MZ, Chebyshev) associated features. We describe an algorithm for generating datasets corresponding to a pre-specified best-fit surface and form error. We use a Gaussian process model to generate form errors that are spatially correlated. The form errors can be drawn from multivariate Gaussian or rectangular distributions. For the latter case a Gaussian copula is used to construct a multivariate multivariate distribution with the pre-assigned correlation. We illustrate the data generation on MZ circle fitting problems. We also describe an approximate MZ circle fitting problem that can be solved using linear programming.

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.

The set of measured data points acquired from the Coordinate Measuring Machine (CMM) need to be processed and analyzed for evaluating the form errors inside the manufactured components. This paper presents a modified algorithm of particle swarm optimization (MPSO) for assessing the form error from the set of coordinate measured data points. In the classical algorithm of the particle swarm optimization (PSO), the value of the candidate solution is updated from its existing value without actually comparing the value obtained in the consecutive iterations for fitness. This behaviour attributes to a lack of exploitation ability in the defined search space. The proposed algorithm generates new swarm position and fitness solution for the objective function through an improved and modified search equation based on a proposed heuristic step. In this step, the swarm searches around the best solution of the previous iteration for improving the swarm exploitation capability. The particle swarm uses greedy selection procedure to choose the best candidate solution. A non-linear minimum zone objective function is formulated mathematically for different types of form errors and then optimized using proposed MPSO. Five benchmark functions are used to prove the effectiveness of the modified algorithm, which is verified by comparing its solution and convergence with those obtained from the established algorithms namely PSO and genetic algorithm (GA). Finally, the result of the proposed algorithm for form error evaluation is compared with previous work and other established nature-inspired algorithms. The results demonstrate that the proposed MPSO algorithm is more efficient and accurate than the other conventional heuristic optimization algorithms. Furthermore, it is well suited for form error evaluation using CMM acquired data.

In practice, it is necessary to understand the general quality status of a batch of circular features machined under the same or similar conditions. In particular, it is necessary to understand the possible worst case among such a batch of circular features. To this end, a brand-new concept called “batch circular form error” is presented in this paper. Detailed definitions of batch circular form error including batch circular form error with ideal conditions and batch circular form error with realistic conditions are provided. Accordingly, the evaluation algorithms of batch circular form error are developed. The algorithms include the characterization of the deterministic profile of circular features and the evaluation of batch circular form error based on a given profile confidence level. Case studies with simulation and experimental data are used for demonstration. The results show that the batch circular form error can be estimated with the data measured from any circular feature in the batch, as long as the sample size is large enough and random ratio is not too large.

More and more research works on assembly tolerance simulation is considering not only orientation and position deviations, but also form errors. In this paper, we describe a process to generate skin model shapes, i.e. 3D models with form error, from the CAD model of a part. We study several different methods used for generating discrete models with defects, and these methods are applied to an identical CAD model to point out their differences. The work will be of benefit for choosing and applying of these different generation approaches, and for automating the generation of skin model shapes.

Computer-aided tolerancing (CAT) aims to predict and control geometrical and dimensional deviations in the early design stage. Former simulation models based on the translation and rotation of nominal features cannot fulfill engineering demands or cover the product lifecycle. Nonideal feature-based simulation methods are, therefore, drawing a great deal of research attention. Two general problems for non-ideal feature-based methods are how to simulate manufacturing defects and how to integrate these defects into tolerance analysis. In this paper, we focus on the first problem. There are already many manufacturing defect simulation methods. Although they are derived from different fields and have different names, they share common characteristics in application. In this study, we collected different simulation methods and classified them as random noise methods, mesh morphing methods, and mode-based methods. The theoretical backgrounds of these methods are introduced, and the simulation examples are conducted on a consistency model to show their differences. Criteria such as multiscale, surface complexity, measurement data integration, parametric control, and calculation complexity are proposed to compare these methods. Based on these analyses, the advantages and drawbacks of each method are pointed out, which may help researchers and engineers to choose suitable methods for their work.

Linear regression is a basic tool in mobile robotics, since it enables accurate estimation of straight lines from range-bearing scans or in digital images, which is a prerequisite for reliable data association and sensor fusing in the context of feature-based SLAM. This paper discusses, extends and compares existing algorithms for line fitting applicable also in case of strong covariances between the coordinates at each single data point, which must not be neglected if range-bearing sensors are used. Besides, particularly the determination of the covariance matrix is considered, which is required for stochastic modeling. The main contribution is a new error model of straight lines in closed form for calculating fast and reliably the covariance matrix dependent on just a few comprehensible and easily obtainable parameters. The model can be applied widely in any case when a line is fitted from a number of distinct points also without a-priori knowledge of the specific measurement noise. By means of extensive simulations the performance and robustness of the new model in comparison to existing approaches is shown.

Industrial parts are routinely affected by dimensional and geometric errors originated in the course of manufacturing processes. These errors, whose pattern is typically related to a specific machining or forming process, are controlled in terms of dimensional and geometrical tolerances (such as e.g. straightness, roundness, flatness, profile) that require verification. In the last two decades, the Kriging model has been put forward for predicting errors of surfaces, whose tolerances are verified using coordinate measuring machines (CMMs), commonly used for 3D measurement on account of both accuracy and flexibility. Kriging is stochastic linear interpolation technique based on an evaluation of how the variance between responses at different points depends on the distance between the two locations. This can be expressed as a variogram which shows how the average difference between values at points changes; it is a function of the distance and of the corresponding direction of any pair of points depicting their correlation extent. The use of the variogram for identifying the correlation structure is recommended by the geostatisticians even if the stochastic process is not stationary. In this paper we resort to empirical variograms to detect possible manufacturing signatures, i.e. systematic pattern that characterises all the features manufactured with a particular production process, and systematic errors of the CMM measurement process. We simulate planar surfaces affected by different and typical manufacturing signatures and possible errors of a measurement process. The behaviour of the variogram gives evidence of spatial correlations, enlightening possible non isotropy. More specifically, we recommend the variograms in different directions of the axis because they draw attention to trend components with a specified direction.

Nowadays freeform components are more and more commonly employed in precision engineering and it is crucial to evaluate their form qualities. The widely adopted PV parameters for form errors need to be evaluated in the sense of minimum zone, which is a non-differentiable optimization problem, and very difficult to be solved. Currently NURBS has become a ubiquitous data format in computer aided design and manufacturing. However, little can be found concerning the minimum zone fitting between the NURBS surfaces and measured data, due to the mathematical complexity of this problem. A fast evaluation strategy is proposed in this paper. It is implemented in three stages. The NURBS model is decomposed into Bezier patches first and the iterative closest point matching is implemented. And then the relative position is refined by the orthogonal distance least squares fitting. Finally the minimum zone fitting is carried out by the primal-dual interior point method. The solutions are recursively updated by line search until the Karush-Kuhn-Tucker conditions are satisfied. Numerical experiments proved that the proposed method is capable of minimum zone fitting for freeform NURBS surfaces with very high accuracy and efficiency.

To address the calculation of elliptic sphere form error, a differential geometry algorithm is proposed and the presented model is answered by fminsearch function in Matlab. An example shows the effectiveness of the proposed algorithm.

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.

The minimum zone method is currently a traditional means for evaluating form error. However, it cannot reveal the effects of form error on the scale of assembly accuracy. To address this deficiency, however, considering the propagation of geometric error in assembly process, this article proposed an entropy-based evaluation method for plane form error for precision mechanical system, which includes primary evaluation, convex hull search, top plane establishment, local entropy analysis and obtaining comprehensive evaluation index. Experimental results demonstrate that the proposed method can evaluate the plane form distribution error, which suggests the assemblability of parts, with the same form errors accurately and stably.

The straightness and flatness errors are generally assessed by using the Least Squares Method (LSM). However, the results obtained from LSM often overestimate the tolerances, and are not consistent with the ISO standards’ definitions. To this end, this paper presents a method to evaluate those errors by using particle swarm optimization (PSO). The realization technique is detailed. The experimental data is utilized to verify this algorithm, together with a comparison with some typical optimization algorithms.

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.

In this paper, we describe techniques for evaluating the uncertainties associated with the assessment of form error, i.e., the departure from ideal geometry of a manufactured part, in coordinate metrology. The techniques take into account measurement uncertainty, sampling effects due to the fact that the form error is determined from a finite set of coordinate data points, and the spatial correlation of the form errors. The techniques are designed to be practical, without the need for complex computation.

Error separation is a key technology for online measuring spindle radial error motion or artifact form error, such as roundness and cylindricity. Three time-domain three-point error separation methods are proposed based on solving the minimum norm solution of the linear equations. Three laser displacement sensors are used to collect a set of discrete measurements recorded, by which a group of linear measurement equations is derived according to the criterion of prior separation form (PSF), prior separation spindle error motion (PSM) or synchronous separation both form and spindle error motion (SSFM). The work discussed the correlations between the angles of three sensors in measuring system, rank of coefficient matrix in the measurement equations and harmonics distortions in the separation results, revealed the regularities of the first order harmonics distortion and recommended the applicable situation of the each method. Theoretical research and large simulations show that SSFM is the more precision method because of the lower distortion.

An extracting method of normal residual error from a ground profile was proposed in this paper. Original measurement data was re-processed for obtaining an actual ground profile in an on-machine contact measurement system. The robust Gaussian regression filtering and NURBS fitting methods were employed to filter and fit these discrete data for getting the accurate realistic ground profile. By utilizing iterative partitioning algorithm, the normal residual error could be accurately obtained between the actual ground profile and the ideal one. Measurement data achieved from grinding experiment of aspheric mould were dealt with. After form error compensation grinding, the difference between normal residual error and original reached 40 nm in stead of initially 140 nm.

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.

An accurate spiral tool path generation method of ultraprecision three-axis turning free form surface is proposed based on symbolic computation in this paper. Many analytic optical free form surfaces often need to be machined to submicron in form error, such as optical nonaxisymmetric aspheric surfaces, but current mainstream CAM systems usually use nonuniform rational basis spline (NURBS) to describe the designed surface and generate tool path. If we want to use these systems, the analytical optical surfaces must be approximated using NURBS surfaces, but it will introduce approximation error and may be difficult to achieve the approximation error less than submicron. More importantly, there is no effective tool path generation method for the special three-axis turning machine tool in current mainstream CAM systems. In this context, we propose to calculate the tool path directly from these analytic surfaces by using symbolic math. The proposed method can be used to generate accurate spiral tool paths for zero/negative/positive rake angle in a unified way. Finally, several examples are given to prove its effectiveness.

An approximate algorithm of exposure distribution on photoresist for the case of large exposure dose is obtained on the basis of the algorithm reported in [X. Dong, C. Du, S. Li, C. Wang, Y. Fu, Control approach for form accuracy of microlenses with continuous relief, Opt. Express 13 (2005) 1353-1360] via analyzing a PAC concentration distribution inside the photoresist. We analyzed the fabrication errors of the micro-optical structures which are caused by the approximate algorithm. The relief form error originating from the measurement errors of characteristics parameters of the photoresist is analyzed as well. A valid approach for a quick and accurate design of the exposure distribution is provided accordingly. As an example, a saw-tooth grating with 75 mu m relief depth was designed and fabricated so as to verify the new algorithm. Our measurement results show that root mean square (rms) value is less than 0.81 mu m. The quick algorithm can satisfy the requirements of micro-optical systems for practical applications. (C) 2009 Elsevier GmbH. All rights reserved.

Dimensional inspection, in integrated manufacturing environments, requires accurate inspection while minimizing the cost and time of inspection. The selection of sampling plan—sample size and sample point locations, the method of evaluating the form error and the nature of the manufactured surfaces will play an important role in deciding the best inspection strategy to be adopted. This paper deals with the strategy for evaluation of flatness error which is one of the most commonly used form tolerances for control of manufactured surfaces. Investigations have been carried out to ascertain the influence of surface quality (surface roughness) in determining the sampling strategy for accurate determination of flatness error while inspecting on a coordinate measuring machine (CMM). The sampling plan utilizes the Hammersley sequence for point location and the flatness error is evaluated using the minimum zone method (MZM) based on computational geometry techniques. Results indicate that the surface roughness influences the accuracy of inspection and can be used as a parameter for determining an initial sample size for the determination of flatness error.

The circular test provides a rapid and efficient way of measuring the contouring accuracy of a machine tool. To get the actual point coordinate in the work plane, an improved measurement instrument - a new ball bar test system - is presented in this paper to identify both the radial error and the rotation angle error when the machine is manipulated to move in circular traces. Based on the measured circular error, a combination of Fourier components is chosen to represent the systematic form error that fluctuates in the radial direction. The typical motion errors represented by the corresponding Fourier components can thus be identified. The values for machine compensation can be calculated and adjusted until the desired results are achieved.

The form error estimation under various machining conditions is an essential step in the assessment of product surface quality generated in machining processes. Coordinate measuring machines (CMMs) are widely used to measure complicated surface form error. However, considering measurement cost, only a few measurement points are collected offline by a CMM for a part surface. Therefore, spatial statistics is adopted to interpolate more points for more accurate form error estimation. It is of great significance to decrease the deviation between the interpolated height value and the real one. Compared to univariate spatial statistics, only concerning spatial correlation of height value, this paper presents a method based on multivariate spatial statistics, co-Kriging (CK), to estimate surface form error not only concerning spatial correlation but also concerning the influence of machining conditions. This method can reconstruct a more accurate part surface and make the estimation deviation smaller. It characterizes the spatial correlation of machining errors by variogram and cross-variogram, and it is implemented on one of the common features: flatness error. Simulated datasets as well as actual CMM data are applied to demonstrate the improvement achieved by the proposed multivariate spatial statistics method over the univariate method and other interpolation methods.

Manufacturing optical components relies on good measurements and specifications. One of the most precise measurements routinely required is the form accuracy. In practice, form deviation from the ideal surface is effectively low frequency errors, where the form error most often accounts for no more than a few undulations across a surface. These types of errors are measured in a variety of ways including interferometry and tactile methods like profilometry, with the latter often being employed for aspheres and general surface shapes such as freeforms. This paper provides a basis for a correct description of power and radius of curvature tolerances, including best practices and calculating the power value with respect to the radius deviation (and vice versa) of the surface form. A consistent definition of the sagitta is presented, along with different cases in manufacturing that are of interest to fabricators and designers. The results make clear how the definitions and results should be documented, for all measurement setups. Relationships between power and radius of curvature are shown that allow specifying the preferred metric based on final accuracy and measurement method. Results shown include all necessary equations for conversion to give optical designers and manufacturers a consistent and robust basis for decision-making. The paper also gives guidance on preferred methods for different scenarios for surface types, accuracy required, and metrology methods employed.

The closed-form solutions for error rates of Space-Time Block Code (STBC) Multiple Phase Shift Keying (MPSK) systems are derived in this paper. With characteristic function based method and the partial integration based respectively, the exact expressions of error rates are obtained for (2,1) STBC with and without channel estimation error. Simulations show that the practical error rates accord with the theoretical ones, so closed-form error rates are accurate references for STBC performance evaluation. For the error of pilot assisted channel estimation, the performance of a (2,1) STBC system is deteriorated about 3dB.

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.

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.

A discrete-cosine-transformation (DCT) based decomposition method is proposed for modeling part form error, which decomposes the error field into a series of independent error modes. Compression, which ensures a compact model, is achieved by correlation reduction or mode truncation based on good energy compaction property of DCT. The part error related to assembly process (rigid body modes) and part distortion during manufacturing (deformation modes) can be separated and identified. DCT has been proven to be equivalent to least square regression with 2D cosine-base for modeling of part variation pattern. Estimation of these parameters in the model has also been developed. The proposed method was applied to model and evaluate assembly and stamping errors at one of the US stamping plant.

Nontraditional functional measures derived from the zone distance measure and associated with form error are computed using a spreadsheet formulation. The functional measures seek to extricate the internal structure of the sample data and include the directional density function and related moments, directional distribution function, balance functional, and a touching point functional. A summary description of the spreadsheet implementation is provided and a detailed example is used to illustrate the geometric structure of the induced measures.

In the process of model validation, models are often declared valid when the differences between model predictions and experimental data sets are satisfactorily small. However, little consideration is given to the effectiveness of a model using parameters that deviate slightly from those that were fitted to data, such as a higher load level. Furthermore, few means exist to compare and choose between two or more models that reproduce data equally well. These issues can be addressed by analyzing model form error, which is the error associated with the differences between the physical phenomena captured by models and that of the real system. This report presents a new quantitative method for model form error analysis and applies it to data taken from experiments on tape joint bending vibrations. Two models for the tape joint system are compared, and suggestions for future improvements to the method are given. As the available data set is too small to draw any statistical conclusions, the focus of this paper is the development of a methodology that can be applied to general problems.

Form error estimation is an essential step in the assessment of product geometry created through one or more manufacturing processes. We present a new method using spatial statistics to estimate form error. Using large sets of uniform sample points measured from five common machined surfaces, we compare the form error estimates using individual points and fitted surfaces obtained through spatial statistical methods. The results show that spatial statistics can provide more accurate estimates of form error under certain conditions. [S1087-1357(00)01701-9]

During form error evaluation, as the accuracy requirement increases the measurement sample size should also be increased, thereby increasing the measurement time. In this paper, a new parameter is introduced based on the asymptotic distribution of the range of errors. This new parameter gives the probability that the form error (range of errors) is less than a predicted value based on the current sample size and the corresponding errors. Simulation studies are carried out and the results are presented.

SUMMARY A method is proposed for comparing configurations of complex machines based on the design features that affect the relative geometric positions of functional elements. Complexity is defined as the level of difficulty associated with securing functional elements relative to one another. The evaluation method focuses on the major components of complexity that govern the relative positions of critical functional elements—connection system error, linear dimension variation and form error. Using the method, the complexity arising from various design decisions is clearly identified by complexity factors obtained for each contributor to positional error in the critical geometric relationships in a machine. It is intended that these simple measures of complexity will be applied at the configuration design stage of the product development process to identify areas of weakness and to enable the selection of the least complex configuration.

Most of the current methods of accurate surface form error measurement is only suitable for linear and regular surface, there is no available accurate method for non-linear surface. This paper fund an online surface form error measurement system based on Time domain least-square serial two-point method which been upgrade by self-adaptable error separation matrix technique for uplift its ability to measure non-linear surface. The self-adaptable error separation matrix technique is allowed adjustment being placed automatically to the error separation matrix for non-linear surface error separation without compromise the accuracy of the system. The comparison of the result from the simulation which was being run on a non-linear surface of 10*9 grids clearly shows that the error's offset of the method in this paper is not larger than the random measurement error. Experimental results show that this method can work on the measurement of non-continuous, irregular surface without accuracy reduction.

In coordinate metrology a common task is to align the frame of reference of the workpiece, as represented by the coordinates gathered by a coordinate measuring system, with that associated with its specification. The positions of geometric features of a work piece are often specified relative to datum features that define an axis system or frame of reference for the work piece. for an ideal geometry and ideal measuring system, subsequent calculations will not depend on which datum features are used to define the frame of reference. For real geometries, the presence of form error associated with the datum surfaces means that different measurement strategies will lead to different definitions of the frames of reference and therefore different estimates of the positions of other geometric features associated with the work piece. In this paper we define approaches for evaluating the contribution of form error associated with datum features to the uncertainties associated with geometric features derived from coordinate data.

Over the last few years considerable attention has been focused on the form fitting algorithms used in coordinate measuring machine (CMM) software. As a result many new algorithms have been developed. However, their accuracy and applicability have been a subject of discussion and argument. This paper describes a research project aimed at a selection process for the most suitable algorithm for evaluating a given form error, and to improve the accuracy of the result using an error compensation technique.

this paper analyzes the performance of fixed decode-and-forward (FDF) relaying and selection decode-and-forward (SDF) relaying network over non-identical Nakagami-m fading channels. The colsed-form outage probabilities and average symbol error probabilities (ASEPs) are derived. Additionally, we provide closed-form outage probability in multiuser scenarios with maximum SNR schedule. These proposed theoretical results are verified by computer simulations.

Radial error measurement of a high speed spindle is commonly affected by speed variations, centring error, form error of master cylinder and thermal drift of the spindle. This paper presents a least square curve fitting technique for processing the high speed spindle data using a mathematical model consisting of a second-order polynomial and sum of sinusoidal functions. Re-sampling method is proposed for accounting the spindle speed variations. Performance of the proposed method is compared with Fourier transform based frequency domain filtering method. Simulation and experimental results confirm the accuracy of the proposed method for analysing the time sampled spindle data. [Received 18 March 2010; Revised 4 October 2010; Accepted 20 October 2010]

International audience; Manufacturers need precise tools to simulate, validate, or improve a process plan for given tolerances. Some simulation methods calculating position and orientation defect of manufactured surfaces have already been developed. A lack in these methods is the integration of form defect of surfaces. Indeed, many methods do not study manufactured surfaces, but nominal models associated to these surfaces. The method developed in this article proposes a tool describing precisely form error in order to take it into account. The work is based on a method of the literature, using discrete cosine transformation, completed by a method for identification of classical defects composing global form error and quantification of their contribution to this defect. The method is validated on simulation examples and then applied on a milled plane.

When dealing with precision in tolerancing of assembly systems, the modelling complexity of the mechanism increases. At first, one can distinguish the 1D tolerancing approach that only concerns variations of dimension. Then, several models are defined to set 3D tolerances, considering that the form error is negligible compared to the orientational and translational variations. Finally, some approaches are proposed to take into account the form variations in the tolerancing of mechanisms. However, some modelling approaches considers the form error as a tolerance zone to add to the 3D tolerances as defined by Rule#l of the ASME standard, or ISO 8015. This paper proposes another point of view, considering the positioning of parts through contact points of their rigid deviation shapes under a defined assembly force and set-up. Rather than considering the positioning of a single part, here is proposed an approach of batch parts assembly by a statistical description of shapes. The result of the method is a statistical positioning error of one part on the other considering the form deviations of parts.

A new concept referred to as the batch roundness is presented in this paper. This concept is derived to illustrate the worst possible form error for a batch of circular features machined under the same conditions. It is to be used as a statistical quality measure for such a batch of circular features and evaluated by analyzing their systematic and random form error components. The definition of batch roundness is introduced first and the associated evaluation algorithm is then presented. The evaluation algorithm starts by characterizing the deterministic profile for the batch of circular features. When the deterministic profile is obtained, the residuals, which are regarded as the random form error component are available. The batch roundness can then be evaluated and the corresponding confidence level of the batch roundness zone determined. Case studies using both the simulated and experimental data sets have successfully demonstrated that the batch roundness can be reliably estimated from the inspection data of only one circular feature in the batch. This unique feature of the presented algorithm will hold as long as the measurement data size is adequate and the relative magnitude of the random form error component with respect to the batch roundness is not too large.

This article puts forward a new measurement of departures from straightness, flatness, roundness and cylindricity in order to conquer the defect of current method on efficiency and precision of data processing, an optimized mathematical model for departure calculation is established respectively. The value of form error is calculated rapidly and precisely on microcomputer by the powerful and optimized calculation function of MATLAB. The solution of departure is established according to minimum zone. It is applied to data processing for dense measurement point especially.

The coordinate measuring machines (CMMs) have proven to be reliable, flexible and very much suitable for determining the acceptability of manufactured parts. In this paper, techniques for evaluating circularity and sphericity error from CMM data are presented. The form error can be evaluated directly from CMM data by employing circle/sphere as assessment features and using normal deviations. The CMM data can also be transformed by applying appropriate methods that not only suppress the size but also introduce distortion. The form error is evaluated from the transformed data by employing limacon/limacoid as assessment features and using linear deviations. The methods for handling CMM and transformed data are given in this paper. The proposed methods are validated using the data available in literature. ? 2003 Elsevier Science B.V. All rights reserved.

The optimal multiuser detector requires extremely high computational complexity. Sub-optimal multiuser detectors can alleviate the problem but still require considerable computational complexity. In this paper, we consider incorporating a multibeam antenna into the multiuser detector to effectively reduce the computational complexity. A method that is well suited to multibeam multiuser detection which we term as "partial decorrelation" is proposed. Closed form error of probability as well as analytical and numerical results of this partial decorrelation detector are given.

The Coordinate Measuring Machines capable of automated measurement are being widely used in on-line and off-line inspection. The measurement data obtained from these inspection devices have to be further processed and analyzed, to assess the geometric form of the manufactured components. In the present work, algorithms based on Computational Geometric techniques have been developed for minimum zone and function-oriented evaluation of straightness and flatness. The algorithms are validated using the simulated data and the data available in the literature.