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Chebyshev reference software

Objective


Chebyshev software has been written to deal with minimum zone (MZ), minimum circumscribed (MC) and maximum inscribed (MI) substitute elements for use in the assessment of form deviation in coordinate measuring machines. This work has shown that these Chebyshev problems are indeed capable of sound solution. In order to qualify as reference software, it was necessary to base the approach on the following considerations: a sound mathematical basis, good algorithm design, carefully engineered software implementations, and thorough testing.

The mathematical basis depends on 2 major underlying principles: the adaptation of mathematical optimization theory, and the use of combinatorics. The former provided valuable solution characterizations in a number of cases, and strategies for proceeding systematically to a solution. The latter provided a guarantee in several cases that the solution obtained is globally best and in addition a subset of the measurement data sets that for the solution of the Chebyshev problem can be regarded as conveying the same information as the complete set.

For many of the geometric elements there exist data sets for which there may be more than one solution. Usually, it is the global solution that is sought, ie that with least Chebyshev error or smallest maximum deviation from the measurement data. For a number of the elements, the global solution(s) is provided. For the remainder, the solution returned will be locally optimal. In the latter cases, the user can employ his knowledge of the workpiece in order to provide an initial estimate that should improve his expectation of obtaining the solution he requires. Automatic initial estimation, which can be overridden by the user, has been provided in many cases.

The algorithm design and software development followed closely the underlying mathematical principles. Each algorithm is as modular as possible in order that modules are reusable across the suite of algorithms. Reliability and robust ness were given high priority. Efficiency was regarded as secondary to the aims of reference software. However, attention was paid to speed of computation where doing so did not compromize the major objectives.

Testing was regarded as a vital part of the work. It was carried out using the BCR data sets and large numbers of sets of simulated data including extreme cases. Computed results where wherever possible assessed by examining whether all the mathematical conditions for a solution held. Indeed, much of the software is self validating in the sense that it is designed to progress systematically from an initial estimate to such a solution. Program 'walk throughs' were carried out to detect inconsistencies and to improve clarity of code. All documentation was carefully checked.

Independent testing was undertaken by a partner who was not directly involved in the development. By using a different computer system, it assisted in making the software more portable. By applying, in addition to the data sets already used, many other data sets generated independently, much valuable information was fed back to improve the software. Improvements also resulted from detailed scrutiny of the supporting documentation. The software is presented mainly in Matlab and TurboPascal; a few public domain Fortran subroutines are called.
Coordinate measuring machines are the heart of the metrological control system flexible manufacturing units. For their operation they rely upon computer software to convert the point information gathered by scanning the object into a 3 dimensional representation of the article being measured. This software is generally based upon a mathematical method called 'least mean square statistics'. However, for many production purposes where tolerances have to be observed, the use of a different mathematical approach based upon "Chebyshev statistics" is more appropriate.
Experience obtained in a current BCR project on the comparison of least mean squares software has shown that much of that currently available is inadequate and non-validated. The purpose of the present project is to develop reference algorithms and software based on the Chebyshev principle against which the software of 3-coordinate measuring machines can be validated. Finally the algorithms and their documentation will be made available to European industry via the metrology laboratories of the Member States thereby assisting it to maintain its world leadership in the field of coordinate measurement technology.
After a careful study of available literature, algorithms will be developed for various basic geometrical forms such as lines, circles, spheres, cylinders and cones or parts thereof. They will be validated using a variety of data sets representing the form under consideration distorted by known amounts. Meetings with representatives of the European coordinate measuring machine industry will be held in the course and at the end of the project to assess the suitability and acceptability of the approach proposed and algorithms developed.

Status:
In progress.

Funding Scheme

CSC - Cost-sharing contracts

Coordinator

National Physical Laboratory (NPL)
Address
Queen's Road
TW11 0LW Teddington
United Kingdom

Participants (2)

Centrum voor Wiskunde en Informatica
Netherlands
PHYSIKALISCH-TECHNISCHE BUNDESANSTALT
Germany
Address
Bundesallee 100
Braunschweig