The Finite Termination Property of an Algorithm for Solving the Minimum Circumscribed Ball Problem

• Autorzy: Gläser A. , Grossmann C. , Lunze U.
• Języki publikacji: EN

Abstrakty

EN

In this paper basic mathematical tasks of coordinate measurement are briefly described and a modified optimization algorithm is proposed. Coordinate measurement devices generate huge data set and require adapted methods to solve related mathematical problems in real time. The proposed algorithm possesses a simplified step size rule and finds the solution of the minimum circumscribed ball fitting after only a finite number The iteration is of the steepest descent type applied to the related distance function. But, in contrast to standard algorithms it uses a modified step size rule that takes into account the specific properties of the occurring objective function. This small difference in the code improves the performance of the algorithm and it enables real time use of the proposed method in coordinate measurement machines. The effciency of the prosed algorithm will be illustrated by some typical examples.

Słowa kluczowe

EN

minimum circumscribed; maximum inscribed; minimum zone; geometrical form elements; coordinate measuring; mini-max problem; steepest descent.

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Schedae Informaticae
• Rocznik: 2012
• Tom: Vol. 21
• Strony: 127--139
• Opis fizyczny: Bibliogr. 15 poz., rys.

Twórcy

autor

Gläser A.

• Institut für Numerische Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany

autor

Grossmann C.

• Institut für Numerische Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany
• Institut für Produktionstechnik, Westsächsische Hochschule Zwickau, D-08056 Zwickau, Germany

autor

Lunze U.

• Institut für Produktionstechnik, Westsächsische Hochschule Zwickau, D-08056 Zwickau, Germany

Bibliografia

• [1] Anthony G.T., Bittner B., Butler B.P., Cox M.G., Drieschner R., Elligsen R., Forbes A.B., Gross H., Hannaby S.A., Harris P.M., Kok J.; Chebyshev reference software for the evaluation of coordinate measureing machine data, NPL Teddington - UK, PTB Braunschweig { Germany, CWI Amsterdam - Netherlands, 1993.
• [2] ASME Y14.5-2009; Dimensioning and Tolerancing, American Society and Mechanical Engineers, 2009.
• [3] Bobrow J.E.; A Direct Minimization Approach for Obtaining the Distance betweenConvex Polyhedra, Int. J. of Robotics Res. 11, 1992, p. 5.
• [4] BS 7172; British Standard guide to assessment of position, size and departure from nominal form of geometric feature.
• [5] Christoph R., Neumann H.J.; Multisensor-Koordinatenmesstechnik, sv corporate media, 2006.
• [6] Dem'janov W.F., Malozemov W.N.; Introduction to Minimax, Dover Publication, New York 1990.
• [7] Grossmann C., Kaplan A.A.; Strafmethoden und modifizierte Lagrange Funktionen in der nichtlinearen Optimierung, Teubner, Leipzig 1979.
• [8] Grossmann C., Kleinmichel H., Vetters K.; Minimaxprobleme und nichtlineare Optimierung, Math. Meth. OR 20, 1976, pp. 23{38.
• [9] Goch G., Lübke K.; Tschebyscheff approximation for the calculation of maximum inscribed/minimum circumscribed geometry elements and form deviations, CIRP Annals - Manufacturing Technology 57 I. 1, 2008, pp. 517-520.
• [10] Hadrich M.; Ein Verfahren zur Tschebyscheff-Approximation von Formelementen der Koordinatenmesstechnik, Measurement, vol. 12, 2004, pp. 337-344.
• [11] ISO 1101; Geometrical product specification (GPS) - geometrical tolerating - tolerating of form, direction, position und run, Beuth Verlag, Berlin 2006.
• [12] Koch W., Lotze W., Lunze U.; Paarungslehrung nach dem Taylorschen Grundsatz durch nichtlineare Optimierung { Praxiseinsatz in der modernen Fertigungstechnik, QZ Qualität und Zuverlässigkeit 36(4), 1991, pp. 219-224.
• [13] Lotze W.; General solution for Tsebyshev approximation of forn elements in coordinate measurement, Measurement 12, 1994, pp. 337-344.
• [14] The MultiSensor; The International Newsletter of Werth Messtechnik. http://www.werth.de/en/navigation/werth-newsletter.html
• [15] Späth H., Watson G.A.; Smallest circlumscribed, largest inscribed and minimum zone circle or sphere via sequential linear programming, Math. Comm. 6, 2001, pp. 29-38.