Relative position computation for assembly planning with planar toleranced parts

In this paper we present a new framework for worst-case toleranced assembly planning of planar mechanical systems. Unlike most assembly planners, which produce plans for nominal parts, our framework incorporates the inherent imprecision of the manufacturing process, which introduces uncertainties in the shape and size of the assembly parts. It accounts for the uncertainty of the relative part placements in the assembled state and allows planning for all assembly instances. Our framework is more general than existing approaches in terms of the part model, the tolerance specification model, and the type of motions used in the assembly sequences. We describe efficient algorithms for computing the sensitivity of part positioning to variations in part geometries, and for incorporating these computations into the geometric core of existing assembly planners for nominal parts. We show that the cost of accounting for toleranced parts in planning is a multiplicative factor which is a polynomial of low degree in the number of tolerance parameters. Our implementation and experiments on five assembly models show that tolerancing significantly reduces the volume of the space of valid motions for assembly sequence planning, and that for translational motions this reduction depends linearly on the size of the tolerance intervals. We conclude that geometric computation for assembly planning with tolerance parts is time efficient and practical.