Theory and Algorithms for L1 Fitting Used for Planar Datum Establishment in Support of Tolerancing Standards

2013 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Convex Hull ◽  
Reference Frame ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Discrete Point ◽  
L1 Norm ◽  
Point Sampling ◽  
Measuring Machine ◽  
Coordinate Data

We present the theory and algorithms for establishing a datum plane consistent with ASME Y14.5 standard definitions. Such a datum plane will correspond to a planar datum feature sampled with coordinate data that is weighted. The method uses a one-sided minimization search based on the L1 (L1) norm. We prove that the problem reduces to a simple minimization search between the weighted centroid and the convex hull. The practice of unweighted fitting works well enough when point sampling is controlled and can thus be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we show by example that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. Terse Mathematica code is included for the reader. The code is sufficient for constrained and unconstrained planar fitting as well as a 3-2-1 datum reference frame generation, which is also described in detail. We additionally prove convergence from the discrete to continuous cases of datum establishment as the point sampling becomes dense.

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

Theory and Algorithms for Weighted Total Least-Squares Fitting of Lines, Planes, and Parallel Planes to Support Tolerancing Standards

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present the theory and algorithms for fitting a line, a plane, two parallel planes (corresponding to a slot or a slab), or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3 × 3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact coordinate measuring machine). However, we show by example that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

Integrating a Vision System With a Coordinate Measuring Machine to Automate the Datum Alignment Process

2005 ◽  
Author(s):  
Rajesh Subramanian ◽  
H. James de St. Germain ◽  
Samuel Drake
Keyword(s):  
Reference Frame ◽  
Vision System ◽  
Coordinate Measuring Machine ◽  
Inspection Time ◽  
Alignment Procedure ◽  
Machined Parts ◽  
Alignment Technique ◽  
Alignment Process ◽  
Measuring Machine ◽  
Part Fixturing

Inspection is an important stage in the manufacturing process of machined parts. Coordinate measuring machines (CMM) have become more automatic, programmable, and capable of fulfilling the growing demands of inspection. However, fixturing (datum alignment) of parts is still done manually, consuming valuable inspection time. In this paper, we describe an automated datum alignment technique which integrates a vision system with the CMM to avoid part fixturing. The rough position of the part is estimated through image analysis. This initial reference frame drives the CMM through an automatic datum alignment procedure, thereby automatically establishing the reference frame without the use of fixtures. This technique has been demonstrated for two and a half dimensional (2.5D) machined parts with well-defined features that exhibit a stable position on a flat table.

Inspecting Profile Errors of Conjugate Disk Cams With Coordinate Measurement

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Wen-Tung Chang ◽  
Long-Iong Wu ◽  
Kuang-Hua Fuh ◽  
Chen-Chou Lin
Keyword(s):  
Interpolation Method ◽  
Measurement Data ◽  
Coordinate Measuring Machine ◽  
Coordinate Measurement ◽  
Numerical Interpolation ◽  
Radial Dimension ◽  
Cam Profile ◽  
Measuring Machine ◽  
Profile Errors ◽  
Coordinate Data

To inspect cam profile errors, a coordinate measuring machine (CMM) is frequently employed. The coordinate data of discrete points at the cam surface are measured by a CMM in order to further evaluate the actual cam profile with complicated interpolation algorithms, and then the cam contour approximated by these algorithms may be compared with the theoretical one. In this paper, a direct and simple analytical method, instead of a numerical interpolation algorithm, is proposed for inspecting the profile deviations of conjugate disk cams with coordinate measurement data. The method is based on the derived correlation between the radial-dimension errors and the normal-direction errors of conjugate cam profiles. To verify the proposed method, an experiment of inspecting a pair of conjugate disk cam profiles was conducted. The experimental results obtained from the proposed method were compared with those obtained by using the Hermite interpolation method. It shows that this method is accurate and more efficient for inspecting the profile errors of conjugate disk cams.

scholarly journals Evaluation of dimensional accuracy of dental bridges manufactured with conventional casting technique and CAD/CAM system with Ceramill Sintron blocks using CMM

2018 ◽  
Vol 12 (4) ◽  
pp. 264-271 ◽  
Author(s):  
Alireza Izadi ◽  
Fariborz Vafaee ◽  
Arash Shishehian ◽  
Ghodratollah Roshanaei ◽  
Behzad Fathi Afkari
Keyword(s):  
Reference Model ◽  
Coordinate Measuring Machine ◽  
Dimensional Accuracy ◽  
Three Dimensions ◽  
Casting Technique ◽  
Dimensional Changes ◽  
Conventional Casting ◽  
Cad Cam ◽  
Milling Method ◽  
Measuring Machine

Background. Recently, non-presintered chromium-cobalt (Cr-Co) blocks with the commercial name of Ceramill Sintron were introduced to the market. However, comprehensive studies on the dimensional accuracy and fit of multi-unit frameworks made of these blocks using the coordinate measuring machine (CMM) are lacking. This study aimed to assess and compare the dimensional changes and fit of conventional casting and milled frameworks using Ceramill Sintron. Methods. A metal model was designed and scanned and 5-unit frameworks were fabricated using two techniques: (I) the conventional casting method (n=20): the wax model was designed, milled in the CAD/CAM machine, flasked and invested; (II) the milling method using Ceramill Sintron blocks (n=20): the wax patterns of group 1 were used; Ceramill Sintron blocks were milled and sintered. Measurements were made on the original reference model and the fabricated frameworks using the CMM in all the three spatial dimensions, and dimensional changes were recorded in a checklist. Data were analyzed with descriptive statistics, and the two groups were compared using one-way ANOVA and Tukey test (α=0.05). Results. The fabricated frameworks in both groups showed significant dimensional changes in all the three dimensions. Comparison of dimensional changes between the two groups revealed no significant differences (P>0.05) except for transverse changes (arch) that were significantly greater in Ceramill Sintron frameworks (P<0.05). Conclusion. The two manufacturing processes were the same regarding dimensional changes and the magnitude of marginal gaps and both processes resulted in significant dimensional changes in frameworks. Ceramill Sintron frameworks showed significantly greater transverse changes than the conventional frameworks.

scholarly journals The Influence of Automated Machining Strategy on Geometric Deviations of Machined Surfaces

2021 ◽  
Vol 11 (5) ◽  
pp. 2353
Author(s):  
Ján Varga ◽  
Teodor Tóth ◽  
Peter Frankovský ◽  
Ľudmila Dulebová ◽  
Emil Spišák ◽  
...  
Keyword(s):  
Control System ◽  
Surface Quality ◽  
Coordinate Measuring Machine ◽  
Geometric Tolerance ◽  
Geometric Tolerances ◽  
Significant Difference ◽  
Geometric Deviations ◽  
Measuring Machine ◽  
Overlap Parameter ◽  
Total Average

This paper deals with various automated milling strategies and their influence on the accuracy of produced parts. Among the most important factors for surface quality is the automated milling strategy. Milling strategies were generated from two different programs, CAM system SolidCAM, with the help of workshop programming in the control system Heidenhain TNC 426. In the first step, simulations of different toolpaths were conducted. Using geometric tolerance is becoming increasingly important in robotized production, but its proper application requires a deeper understanding. This article presents the measurement of selected planes of robotized production to evaluate their flatness, parallelism and perpendicularity deviations after milling on the coordinate measuring machine Carl Zeiss Contura G2. Total average deviations, including all geometric tolerances, were 0.020 mm for SolidCAM and 0.016 mm for Heidenhain TNC 426. The result is significantly affected by the flatness of measured planes, where the overlap parameter of the tools has a significant impact on the flatness of the surface. With interchangeable cutter plate tools, it is better to use higher overlap to achieve better flatness. There is a significant difference in production time, with SolidCAM 25 min and 30 s, and Heidenhain 48 min and 19 s. In accordance with these findings, the SolidCAM system is more suitable for production.

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