Effect of tolerancing on the relative positions of parts in an assembly

This paper analyzes the variations of the relative positions of the parts composing an assembly when part dimensions span their specified tolerance zones. We focus on the case where each part is modeled as a toleranced polygon defined as follows: each edge of this polygon is supported by a line of fixed orientation lying anywhere in a strip bounded by two extreme lines; these lines are themselves defined in a coordinate system attached to the part. The relative placements of parts in an assembly A are defined by spatial relations, such as edge e of part P is parallel to edge f of part Q, at distance d. We define the relative position of any two parts, P and Q, in an instance of A as the transform (a translation in our case) between the coordinate systems attached to these parts. Because of the possible variations in the geometry of each individual part, the relative position of P and Q is not constant over multiple instances of A. Hence, the question: What is the range of possible relative positions of any two parts in A? This paper describes an efficient divide-and-conquer algorithm to solve this problem. Experimental results obtained with an assembly sequence planner that incorporates this algorithm are also presented.