Qualitative models of 3-D frame-transformation motions

Qualitative spatial reasoning has been used for representing landmarks, path and grasp planning, shape recognition etc, but has generally not been extended beyond 2-D. In this paper, we develop a 3-D spatial reasoning algebra, using qualitative versions of (a) Gibbs-Heaviside vector calculus with homogeneous coordinates, and (b) screw calculus motions that map one object frame onto another. By comparing the results of composition in one, two and three dimensions, we find that the degree of uncertainty is much higher in 3-D. We demonstrate the utility of a qualitative formulation in a simple robot spatial task with imprecise instructions.