Geometric constraint subsets and subgraphs in the analysis of assemblies and mechanisms

The Geometric Reasoning skill is central to many CAD / CAM / CAPP (Computer Aided Design, Manufacturing and Process Planning) applications. There is a growing demand for Geometric Reasoning systems that assess the feasibility of virtual scenes, specified by geometric relationships. Therefore, the problem of Satisfaction of Geometric Restrictions or of Feasibility of Scene (GCS / SF) consists of a basic scenario containing geometric entities, the context of which is used to propose constraint relationships between entities that are still undefined. If the specification of the restrictions is consistent, the answer to the problem is one of the finite or infinite number of solution scenarios that satisfy the proposed restrictions. Otherwise, a diagnosis of inconsistency is expected. The three main strategies used for this problem are: numerical, procedural and mathematical. The numerical and procedural solutions solve only part of the problem, and are not complete in the sense that an absence of response does not mean the absence of it. The mathematical approach previously presented by the authors describes the problem using a series of polynomial equations. The roots common to this set of polynomials characterize the solution space for the problem. This paper presents the use of techniques with Groebner Bases to verify the consistency of the restrictions. She also integrates the subgroups of the special Euclidean group of displacements SE (3) in the formulation of the problem to exploit the structure implied by geometric relationships. Although theoretically sound, these techniques require large amounts of computational resources. This paper proposes Divide and Conquer techniques applied to local GCS / SF subproblems to identify sets of geometric entities strongly restricted to each other. The identification and pre-processing of such local assemblies generally reduces the effort required to solve the entire problem. The identification of these local sub-problems is related to the identification of short cycles in the Geometric Restrictions graph of the GCS / SF problem. Its preprocessing uses the aforementioned techniques of Algebraic Geometry and Groups in the local problems that correspond to these cycles. In addition to improving the efficiency of the solution, the Divide and Conquer techniques capture the physical essence of the problem. This is illustrated by its application to the analysis of degrees of freedom of mechanisms.

• Autores: E. Ruiz, Oscar
• Fecha de publicación: 2006
• Institución: Universidad EAFIT
• Repositorio: Repositorio EAFIT
• Idioma: eng
• OAI Identifier: oai:repository.eafit.edu.co:10784/14565
• Acceso en línea: http://hdl.handle.net/10784/14565
• Palabra clave: Graph Cycle; Groebner Basis; Constraint Graph; Mechanisms; Assemblies; Ciclo De Gráfico; Base De Groebner; Gráfico De Restricción; Mecanismos; Ensamblajes
• License: Copyright (c) 2006 Oscar E. Ruiz