Research Papers

Datum Planes Based on a Constrained L1 Norm

[+] Author and Article Information
Craig M. Shakarji

Physical Measurement Laboratory,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: craig.shakarji@nist.gov

Vijay Srinivasan

Fellow ASME
Engineering Laboratory,
National Institute of Standards and Technology,
Gaithersburg, MD 20899
e-mail: vijay.srinivasan@nist.gov

Contributed by the Manufacturing Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received January 12, 2015; final manuscript received September 28, 2015; published online November 2, 2015. Editor: Bahram Ravani. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Comput. Inf. Sci. Eng 15(4), 041008 (Nov 02, 2015) (8 pages) Paper No: JCISE-15-1017; doi: 10.1115/1.4031827 History: Received January 12, 2015; Revised September 28, 2015

This paper has two major goals. First, we present an algorithm for establishing planar datums suitable for a default in tolerancing standards. The algorithm is based on a constrained minimization search based on the L1 (L1) norm after forming a convex surface from the original surface or sampled points. We prove that the problem reduces to a simple minimization search between the convex surface and its centroid. The data points in the discrete case do not need to have any corresponding weights provided with them, as appropriate weighting is part of the algorithm itself, thereby making the algorithm largely insensitive to nonuniformly sampled data points. Terse mathematica code is included for the reader. The code is sufficient for primary and secondary planar datum fitting as well as a 3-2-1 datum reference frame generation. The second goal of this paper is to compare this new method with several other possible means for establishing datum planes, ultimately showing several appealing characteristics of the proposed algorithm. Since both the International Organization for Standardization (ISO) and American Society of Mechanical Engineers (ASME) standardization efforts are actively working to establish datum plane definitions, the timing of such a study is opportune.

Copyright © 2015 by ASME
Topics: Algorithms
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Srinivasan, V. , 2013, “ Reflections on the Role of Science in the Evolution of Dimensioning and Tolerancing Standards,” Proc. Inst. Mech. Eng., Part B, 227(1), pp. 3–11. [CrossRef]
Tandler, W. , 2008, All Those Datum Things, Quality Digest Magazine, Chico, CA.
Tandler, W. , 2008, Establishing Datum Reference Frames, Quality Digest Magazine, Chico, CA.
ANSI/ASME Y14.5.1M, 2009, Dimensioning and Tolerancing, The American Society of Mechanical Engineers, New York.
ANSI/ASME Y14.5.1M, 1994, Dimensioning and Tolerancing, The American Society of Mechanical Engineers, New York.
ISO 5459, 2011, Geometrical Product Specifications (GPS)—Geometrical Tolerancing—Datums and Datum Systems, International Organization for Standardization, Geneva, Switzerland.
Xuzeng, Z. , and Roy, U. , 1993, “ Criteria for Establishing Datums in Manufactured Parts,” J. Manuf. Syst., 12(1), pp. 36–50. [CrossRef]
Shakarji, C. M. , and Srinivasan, V. , 2013, “ Theory and Algorithms for L1 Fitting Used for Planar Datum Establishment in Support of Tolerancing Standards,” ASME Paper No. DETC2013-12372.
Hopp, T. H. , 1990, “ The Mathematics of Datums,” American Society for Precision Engineering Newsletter, Last accessed Sept. 2015, http://www.mel.nist.gov/msidlibrary/doc/hopp90.pdf
Shakarji, C. M. , 2011, “ Coordinate Measuring System Algorithms and Filters,” Coordinate Measuring Machines and Systems, J. Hocken and P. H. Pereira , eds., CRC Press, Boca Raton, FL, pp. 153–182.
Shakarji, C. M. , and Srinivasan, V. , 2013, “ Theory and Algorithms for Weighted Total Least-Squares Fitting of Lines, Planes, and Parallel Planes to Support Tolerancing Standards,” ASME J. Comput. Inf. Sci. Eng., 13(3), p. 031008. [CrossRef]
Weisstein, E. , “ Convex Hull,” From MathWorld—A Wolfram Web Resource, Last accessed Jan. 2015, http://mathworld.wolfram.com/ConvexHull.html
O'Rourke, J. , 1998, Computational Geometry in C, 2nd ed., Cambridge University Press, Cambridge, UK.
Houle, M. E. , and Toussaint, G. T. , 1988, “ Computing the Width of a Set,” IEEE Trans. Pattern Anal. Mach. Intell., 10(5), pp. 761–765. [CrossRef]
Barber, C. B. , Dobkin, D. P. , and Huhdanpaa, H. , 1996, “ The Quickhull Algorithm for Convex Hulls,” ACM Trans. Math. Software, 22(4), pp. 469–483. [CrossRef]


Grahic Jump Location
Fig. 1

Deriving a datum plane from a datum feature

Grahic Jump Location
Fig. 2

Fitting a plane to a surface patch

Grahic Jump Location
Fig. 5

An example of the lower convex envelope derived from the data points (shown in 2D for simplicity)

Grahic Jump Location
Fig. 6

An example case showing the L∞ -based definitions yielding an undesired slope of zero

Grahic Jump Location
Fig. 7

An example case showing some L2 -based definitions that have an undesired negative slope

Grahic Jump Location
Fig. 8

An example of a wavy datum feature showing various datum definitions



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