Analysing Geometric Obstacles. A Theorem on d-Elements

Abstract

The product geometry is a fundamental constructive property that has a strong impact on the basic design choices of the assembly process: the product assembly flotation and decomposition into assembly units. The assembly process must be mounted so that the previously set components and elements of technological system could not create geometric obstacles for the main and auxiliary working moves. The paper considers mathematical modelling methods of geometric constraints and restrictions in computer-aided design systems. Publications, about computer-aided design propose numerous varieties of the so-called direct modelling method for geometric obstacles. The principle of this method is to verify the intersection of the geometric model of a mobile object with a static fragment when the first moves along the chosen straight –line (most often) trajectory. It turned out that even in the best version, the direct method is computationally very expensive for products of medium complexity, consisting of several dozen components. Therefore, it is important and urgent to determine the minimum number of geometric verifications, the results of which can be used to synthesize the correct design choices: the assembly flotation and product decomposition into assembly units. The paper proposes a theoretical-lattice formalization of the geometric obstacle of the product. It is shown that the aggregate of all constructive fragments that are assembled independently and do not contain geometric obstacles form a closed algebraic structure that is a lattice. A theorem on d-elements is proved. This theorem allows us to solve the problem of geometric obstacle by cost-conscious algebraic methods. The paper offers three ways for lattice generation: analysis of anti-chains top-down, lattice reconstruction using a set of generative elements, and probabilistic conclusion based on the Bayesian networks of confidence.