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Research Papers

On the Enduring Appeal of Least-Squares Fitting in Computational Coordinate Metrology

[+] Author and Article Information
Vijay Srinivasan, Craig M. Shakarji, Edward P. Morse

 Fellow ASME Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899 e-mail: vijay.srinivasan@nist.gov Mem. ASME Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899 e-mail: craig.shakarji@nist.govMem. ASME Department of Mechanical Engineering and Engineering Science,  University of North Carolina at Charlotte, Charlotte, NC 28223 e-mail: emorse@uncc.edu

http://www.nist.gov/pml/div681/grp11/cst-algorithmtesting.cfm

J. Comput. Inf. Sci. Eng 12(1), 011008 (Dec 21, 2011) (15 pages) doi:10.1115/1.3647877 History: Received December 23, 2010; Revised August 02, 2011; Published December 21, 2011; Online December 21, 2011

The vast majority of points collected with coordinate measuring machines are not used in isolation; rather, collections of these points are associated with geometric features through fitting routines. In manufacturing applications, there are two fundamental questions that persist about the efficacy of this fitting—first, do the points collected adequately represent the surface under inspection; and second, does the association of substitute (fitted) geometry with the points meet criteria consistent with the standardized geometric specification of the product. This paper addresses the second question for least-squares fitting both as a historical survey of past and current practices, and as a harbinger of the influence of new specification criteria under consideration for international standardization. It also touches upon a set of new issues posed by the international standardization on the first question as related to sampling and least-squares fitting.

Copyright © 2012 by American Society of Mechanical Engineers
Topics: Fittings
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References

Figures

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Figure 9

Specification of nonuniform profile tolerance. The nominal profile and tolerance zone boundaries will typically be specified in a CAD system, but may be elaborated through basic dimensions on the drawing.

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Figure 10

ASME defines flatness tolerance specification

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Figure 11

ISO defines flatness tolerance specification

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Figure 12

Width of a set of points in a plane

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Figure 13

An open setup to check for flatness using a surface plate and a dial indicator

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Figure 14

Fitting a straight line to a curve in a plane

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Figure 15

Fitting a plane to a surface patch

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Figure 16

Fitting two parallel planes to two surface patches

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Figure 17

(a) Example of nonuniform sampling and its refinement and (b) example of uniform sampling and its refinement

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Figure 18

Convergence using nonuniform discretization

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Figure 7

For the two objective functions shown, it is easier to find the minimum for the one on the left, since it is smoothly varying and since the global minimum is not so much hidden among nearby, local minima

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Figure 8

A set of points taken around the shape shown could have two maximum inscribed circles, one centered at p and one at q. A least-squares fit to the same data would be unique.

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Figure 1

Linear least-squares fitting of a straight line in a plane

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Figure 2

Total least-squares fitting of a straight line in a plane

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Figure 3

Perpendicular distance between a point and a circle in a plane

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Figure 4

Parabolic projection

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Figure 5

Position vector and normal vector at a point on a surface

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Figure 6

A part of a sample test report following the ASME B89.4.10 Standard

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