Sensing Polygon Poses by Inscription

Industrial assembly involves sensing the pose (orientation and position) of a part. Efficient and reliable sensing strategies can be developed for an assembly task if the shape of the part is known in advance. In this paper we investigate the problem of determining the pose of a convex n-gon from a set of m supporting cones, i.e., cones with both sides supporting the polygon. An algorithm with running time O(nm) which almost always reduces to O(n + m log n) is presented to solve for all possible poses of the polygon. As a consequence, the polygon inscription problem of finding all possible poses for a convex n-gon inscribed in another convex m-gon, can be solved within the same asymptotic time bound. We prove that the number of possible poses cannot exceed 6n, given m >= 2 supporting cones with distinct vertices. Experiments demonstrate that two supporting cones are sufficient to determine the real pose of the n-gon in most cases. Our results imply that sensing in practice can be carried out by obtaining viewing angles of a planar part at multiple exterior sites in the plane. As a conclusion, we generalize this and other sensing methods into a scheme named sensing by inscription.