Algebraic geometry and group theory in geometric constraint satisfaction for computer-aided design and assembly planning

Abstract

Mechanical design and assembly planning inherently involve geometric constraint satisfaction or scene feasibility (GCS/SF) problems. Such problems imply the satisfaction of proposed relations placed between undefined geometric entities in a given scenario. If the degrees of freedom remaining in the scene are compatible with the proposed relations or constraints, a set of entities is produced that populate the scenario satisfying the relations. Otherwise, a diagnostic of inconsistency of the problem is emitted. This problem appears in various forms in assembly planning (assembly model generation), process planning, constraint driven design, computer vision, etc. Previous attempts at solution using separate numerical, symbolic or procedural approaches suffer serious shortcomings in characterizing the solution space, in dealing simultaneously with geometric (dimensional) and topological (relational) inconsistencies, and in completely covering the possible physical variations of the problem. This investigation starts by formulating the problem as one of characterizing the solution space of a set of polynomials. By using theories developed in the area of algebraic geometry, properties of Grobner Bases are used to assess the consistency and ambiguity of the given problem and the dimension of its solution space. This method allows for die integration of geometric and topological reasoning. The high computational cost of Grobner Basis construction and the need for a compact and physically meaningful set of variables lead to the integration of known results on group theory. These results allow the characterization of geometric constraints in terms of the subgroups of the Special Group of Euclidean displacements in E ³, SE(3). Several examples arc developed which were solved with computer algebra systems (MAPLE and Mathematica). They are presented to illustrate the use of the Euclidean group-based variables, and to demonstrate the theoretical completeness of the algebraic geometry analysis over the domain of constraints expressible as polynomials.