Evaluation of cylindricity using combinatorics THOM J. HODGSON 1 , MICHAEL G. KAY 1 , RAVI O. MITTAL 2 and SHIH-YI TANG 3 1 Department of Industrial Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA E-mail: hodgson@eos.ncsu.edu 2 Department of Industrial Engineering, North Carolina A&T State University, Greensboro, NC, USA 3 Dimensional Metrology Department, Mail Code: 41622, Cummins Engine Company, Inc., Columbus, IN 47201, USA Several numerical methods have been developed for ®nding the minimum zone of a cylinder. This paper presents a combinatorial method termed the Minimum Shell for cylindricity evaluation. For a given set of measured data points, the method eciently searches for the six points that determine the minimum zone of cylindricity. This method is self-contained, without the requirement of mathematical programming software support, and is equal to or superior to other methods of evaluating cylindricity in terms of accuracy, eciency, and simplicity. Computational results are given. 1. Introduction Dimensional inspection of parts with Coordinate Mea- suring Machines (CMMs) and non-contact technologies has rapidly emerged as an important function in the evolution of computer-integrated manufacturing. As in- dustry is able to realize the usefulness and economic bene®ts of geometric dimensioning and tolerancing, the use of CMMs for inspecting manufactured parts is ex- pected to become even more common. The capability of a CMM to measure parts is a function of many dierent factors, chief among which are the machine's design and construction (hardware), the measurement algorithms (software) used to process collected data, and the in- spection plan (sampling strategy). The discrete data collected by the CMM must be processed by a suitable algorithm in order to estimate the deviation of the actual object from the desired geometry. As dimensional speci®cations become tighter, machine tools become increasingly precise, and part geometry's become more complex, higher standards are demanded of the software used in CMMs. The accuracy of a CMM's output depends on its mechanical construction and the quality of the algorithms used to analyze its data. Man- ufacturers have focused on improving the CMM's me- chanical accuracy, but the algorithms (software) used in CMMs have not received adequate attention [1]. Form tolerances, such as straightness, ¯atness, circu- larity, and cylindricity are important characteristics for manufacturing and assembly. Among these, cylindricity plays an important role in industry because of the central role of bearings and shafts in machines. Design standards for cylindricity can be found in ANSI Y14.5M Dimen- sioning and Tolerancing [2], with its mathematical de®- nition given in ANSI Y14.5.1M Mathematical De®nition of Dimensioning and Tolerancing Principles [3]. Cylin- dricity is more complex than straightness, ¯atness, or circularity, not only because it is three-dimensional, but also because it is de®ned by a mixture of Cartesian and polar coordinate systems [4]. The cylindricity tolerance can be considered as a composite control of form that includes circularity, straightness, and taper of a cylindri- cal feature [2]. No standard criteria for data ®tting exist to determine cylindricity. However, methods for the measurement of roundness [5] can be extended to the measurement of cylindricity. In this paper, an algorithm termed the Minimum Shell method is developed to ®nd the deviation from a set of discrete data measured from CMMs. The algorithm is combinatorial and is based on computational geometry. Experimentally, the results have been shown to be equivalent in quality to the recent Linear Programming based algorithm of Carr and Ferreira [6]. In the next section, de®nitions and terminology used in CMM measurement technology are presented. This is followed by a review of work in measurement algorithms, and a description of the Minimum Shell method. The paper concludes with experimental results, comparing the method to the Carr and Ferreira algorithm. 2. De®nitions and mathematical model 2.1. De®nition of cylindricity The de®nition of cylindricity in ANSI 14.5M [2] is as follows (see also, Fig. 1): 0740-817X Ó 1999 ``IIE'' IIE Transactions (1999) 31, 39±47