INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. X, No. X, pp. X-XX
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DOI: XXX-XXX-XXXX
VECTORIAL METHOD OF MINIMUM ZONE TOLERANCE FOR FLATNESS, STRAIGHTNESS, AND THEIR UNCERTAINTY ESTIMATION Roque Calvo 1#, Emilio Gómez 2, and Rosario Domingo 3 1 Dpt. Mechanical Engineering and Construction. Universidad Politécnica de Madrid. Ronda de Valencia, 3; 28013 Madrid, Spain. #Corresponding author. Phone: +34 913367465. Fax +34 913367676. E-mail:
[email protected] 2 Dpt. Mechanical Engineering and Construction. Universidad Politécnica de Madrid. Ronda de Valencia, 3; 28013 Madrid, Spain. E-mail:
[email protected] 3 Dpt. Construction and Manufacturing Engineering. Universidad Nacional de Educación a Distancia (UNED). Juan del Rosal, 12; 28040 Madrid, Spain. E-mail:
[email protected] KEYWORDS : Flatness, Form tolerance, Minimax problem, Minimum zone, Planar straightness, Measurement uncertainty
Flatness and planar straightness are fundamental form tolerances in engineering design and its materialization through manufacturing processes. Minimum zone tolerance is a preferred approach of flatness and straightness for widely accepted ISO and ANSI standards. In this paper, we propose a novel accurate method of minimum zone tolerance based on vectorial calculus of point coordinates. The non-linear minimax formulation of the original flatness or straightness problem is transformed into a set of linear problems. Next, the optimal solution of the envelop planes or lines is reached through vectorial calculus for both flatness and planar straightness. Then, the developed algorithms are compared to a selection of methods with published tests in recent and classic literature on the topic, reaching the best attained accuracies or outperforming them in the trials. Finally, we propose a new decomposition of the uncertainty contributions for analysis and the improvement of sampling strategy. We conclude remarking the practical contributions of the proposals. Manuscript received: August XX, 201X / Accepted: August XX, 201X
1. Introduction Flatness and straightness are both fundamental tolerances of form in precision design and manufacturing engineering, for product dimensioning and its verification through direct measurement, or as a support to verify other specifications. The tolerances of prismatic parts are ordinary referred to a datum, plane or line, idealization of a physical plane or its orthogonal projection. The measurement of angular magnitudes or squareness is also subject to the determination of planes or lines and their tolerances. In machining, not only the tolerance of the parts, but those of the supporting tooling and the machine tool itself are involved in the manufacturing process capability and its control for specification compliance1. Relative deviations of physical planes or lines are often controlled in the job shop with analogical dial instruments, displacing its probe on the surface, but today digital measurement on coordinate measuring machines (CMM) are widespread for product verification. The CMM uses a set of discrete points coordinates that is evaluated to determine a plane or a line. The standards ISO 11012 or ANSI Y14.53 establish the minimum zone (MZ) as a preferred criterion of tolerance
of form, before other used methods like the least-squares. The standards consider flatness (planar straightness) the minimum distance between two parallel planes (lines) with all the set of points between them -envelop planes or lines-. Even so, the standards do not establish explicitly how to determine the minimum zone. Conversely, the least squares criterion is univocally defined by its mathematical formulation and direct solution. In addition, the least-squares criterion conveys the statistical power of maximum likelihood in the optimization, when the measured point set is treated statistically. The interval of confidence of the model estimators -plane or straight line coefficients- complements the analysis of uncertainty in its statistical approach. Noteworthy, the widely accepted approach to express measurements and their uncertainties lays on the statistical treatment of the variables, where the least-squares criterion finds also its own roots. Therefore, due to several reasons including its easy calculation, least-squares algorithms are generally in the routines of CMM machines. Nevertheless, the physical interference and contact of surfaces are determined by the outstanding points of the plane, so the minimum zone criteria responds better to functional tolerancing4 and fitting.
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INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING
The MZ criterion is referenced in early works by Chetwynd5 in the research of dual conditions of linear programming models. Antony et al.6 confirmed that at least 4 measured points must be included for flatness assessment: In two different possible configurations, 2-2 and 3-1, depending on the minimum number of points lying on each of the two envelop planes. In order to ensure an exact MZ solution, these necessary conditions would require evaluating all possible configurations from a given data set in a combinatorial hard problem. These configurations will be revised later in the construction of the proposed algorithm. According with the ISO or ANSI standards, the determination of the MZ for flatness involves not only the definition of the envelop planes from the data points measured, but also its associated uncertainty. Since the set of points is a sample from the whole surface, the strategy of sampling includes the path of measurement and the number of points. Previous research shows7 that the number of points has a higher influence than the path of sampling in the results uncertainty. Due to the close relationship of the number of measured points with the measuring cost, several works8,9 have sought the reduction of the sample size for a desired precision. This paper is primary dedicated to the process of calculating the flatness or planar straightness and its uncertainty after proper sampling. Looking for an accurate calculation of flatness, several methods have been developed since the 1980’s that can be classified in a first instance in two main categories10: Based and non-based on computational geometry. We can find other classifications11 where computational geometry methods are at the same level of leastsquares methods or recent meta-heuristics. The first taxonomy by Lee10 is preferred and his review is followed and complemented in the next paragraphs, because the research line associated with computational geometry has provided reference values to other new or improved methods. We can mention among the non-computational geometry methods: • Direct least squares12 or weighted least squares13. Its MZ interval is ordinary overestimated, when the data points are not well aligned with a CMM coordinate axis. Shunmugam14 proposed a method based on the median, more robust estimator that the mean. • Non-linear optimization techniques. Its goal is minimizing the maximum distance from an ideal reference plane or point. Some works in the field are those by Shunmugam15, Wang16, Kaiser and Krishnan17, Damodarasamy and Anand18 or Cheraghi et al.19. A characteristic point of these non-linear search methods is the non-convexity of the optimization problem and the need of several trials to look for a global optimum. • Approximation methods. Based on linear programming, in some cases they sacrifice accuracy for an easy implementation. We find in this area the already mentioned work of Chetwynd5. In addition, Prakasvudhisarn20, Car and Ferreira21, Weber et al.22 or Zhu and Ding23. © KSPE and Springer 2011
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• Exchange methods. They construct the solution in a sequential search of a better solution replacing a current set of points by a new one. The works of Fukuda and Shimokohbe24, Huang et al.25, Deng et al.26, Danish and Shunmugam27 or Burdekin and Pahk28 are between them. • Meta-heuristic methods. They have been applied to the minimum zone problem after successful use in difficult combinatory problems. In this category we find the genetic algorithm (GA) of Sharma et al.29, Cui et al.30 or Liu et al.31 combining GA with geometric calculation, particle swarm optimization by Kovvur32, and the gradient ascent approach of the evolution algorithm by Malyscheff et al.33. Some weakness of these methods could be the non-convergence to the exact solution or the computation difficulties facing big data sets. As a main second type of methods, computational geometry tackles the problem through the construction of a convex hull and the identification of the smallest convex domain including all the points. Works in this field includes Houle and Toussaint34, Anthony et al.6, Traband et al.35, Samuel and Shunmugam36, Hermann37 and Lee10. These methods based on convex hull enumerate all the possible solutions looking for an accurate result. For planar straightness many authors have tried to adapt the same spatial algorithm used for flatness, but others have tried to get benefit directly from the 2D geometry, like the iterative method by Danish and Mathew38. In this context, our research in the tolerance of form and fitting is looking for analytic solutions39. We propose a new method that is developed in Section 2, including its application for planar straightness in Section 3. Then, we apply it in Section 4 for performance evaluation to data sets from the literature, focused on the most accurate methods referenced above. Across Section 5 we estimate the uncertainty associated to flatness evaluation, in a breakdown useful for sampling strategy improvement. Finally, in Section 6 we evaluate the results with concluding remarks.
2. Vectorial formulation of the minimum zone for flatness We consider the minimum zone for flatness where the envelop planes must comply with: • At least one point of the data set must lie on each envelop plane. Suppose by the moment that π1 and π2 are the envelop planes of minimum zone at a minimum distance d and they do not contain at least one point of the data set. In this case, we can always find a pair π’1 and π’2 of parallel planes to π1 and π2 at a distance d’
