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Modeling and Measurement Using Implicit Polynomials
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with
William Wolovich
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Hassan Albakri
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One often must compare several (ideally) identical objects to a model object, such as the turbine blades in a jet engine, the propeller blades of a ship, or the contoured grooves in a gear. If the profile and tooth trace errors of a gear exceed tolerance errors, the running qualities of the gear decrease significantly, causing increased tip wear that can seriously degrade overall gear performance. More serious consequences can occur if manufactured objects, such as the turbine blades in a jet engine are not precisely fabricated. Coordinate measuring machines (CMM) are designed to measure complex shapes. |
| They are particularly important when used to quickly and accurately inspect a large variety of manufactured parts, especially those with sculptured, or free-form shapes, such as gears, propeller blades etc., as specified in some quantitative way by an engineer or designer. Master measurements generally are entered into the database of the CMM computer so that it can trace along appropriate profiles and compare its measurement data to the stored data. Here we assume that the objects being traced by the stylus of the CMM machine are modeled by implicit polynomial (IP) equations, which are beginning to play a more important role in manufacturing. We also assume that the stylus of the CMM machine begins its movement very close to some initial point on an IP path to be traced. Subsequent points along the path are then determined using a new perpendicular distance approximation algorithm together with Newton’s root finding procedure. Certain geometrical features of the implicit curve or surface, such as its tangent and curvature are used as well. |
| Implicit polynomial representations are very useful for modeling given point data sets, and numerous papers have been written which illustrate their importance in image understanding and object recognition. Unfortunately, there is very little in the literature on implicit polynomial models for large or entire free-form shapes because of the lack of tractable computational procedures for obtaining and modifying such models. Recently, however, some useful new procedures and computer algorithms have been developed for describing and analyzing them, as well as fitting IP equations to 2-D and 3-D boundary data sets. |
| We developed an algorithm for approximating the perpendicular distance between a point and an implicit polynomial curve. We employed this algorithm in a fast new procedure for defining a sequence of points for a CMM to follow and measure along a curve or surface defined by an IP equation. We illustrated how our new model-based measurement procedure can be used quickly and accurately to determine the distance errors between model shapes and "identical" manufactured shapes. |
| Related Publications |
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"The Precise Modeling and Measurement of Free-Form Surfaces," Transactions of the ASME, Journal of Manufacturing Science and Engineering, 124(2): 326-332, May 2002. [Abstract] [ps] [pdf] |
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"Determining CMM Motion on Free-Form Objects Using Implicit Polynomials," Proceedings of NSF Design, Manufacturing and Industrial Innovation Research Conference, San Juan, Puerto Rico, January 7-10, 2002. [Abstract] [ps], [pdf] |
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"Accurate Implicit Polynomial Measurement Of Sculptured Surfaces," Proceedings of NSF Design, Manufacturing and Industrial Innovation Research Conference, Tampa, Florida, January 7-10, 2001. |
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"The Precise Modeling and Measurement of Sculptured Surfaces," Proceedings of NSF Design, Manufacturing and Industrial Innovation Research Conference, Vancouver, BC, January 3-6, 2000. |
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Carnegie Mellon University, Robotics Institute
5000 Forbes Av., Pittsburgh, PA, 15213
hulyayalcin@gmail.com
