Modeling Geometric Obstacles in the Assembly of Complex Products

Abstract

Geometric properties of products have a great influence on the process of assembly. The problem of modeling geometric obstacles in CAD systems is considered. A review of modern methods for modeling geometric obstacles is performed. It is shown that these methods do not allow minimizing the number of geometric tests. Therefore, they require high computational costs. A concept of g-situation is introduced. G-situation is a mathematical description of assembled product fragments for which a test for geometric obstacles is correct and necessary. Two assertions about geometric inheritance during assembly are formulated. A mathematical model of geometric solvability in the assembly of complex products is proposed. It is a game of the decision maker and nature by coloring the vertices of an ordered set. A rational game strategy allows you to minimize the number of necessary geometric tests that are performed using motion planning or collision detection algorithms. To minimize the number of tests, various a priori and a posteriori algorithms of coloring ordered sets can be applied. A theorem on the properties of colored ordered sets is proved.