The Complexity of Sensing by Point Sampling

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 two problems of determining the pose of a polygonal part of known shape, for the cases of a continuum and a finite number of possible poses, respectively. The first problem, named sensing by inscription, involves determining the pose of a convex n-gon from a set of m supporting cones. 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. We prove that the number of possible poses cannot exceed 6n, given m >= 2 supporting cones with distinct vertices. Simulation 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. On many occasions a parts feeder will have reduced the number of possible poses of a part to a small finite set. Our second problem, named sensing by point sampling, is concerned with a more general version---finding the minimum number of sensing points required to distinguish between n polygonal shapes with a total of m edges. In practice this can be implemented by embedding a series of point light detectors in a feeder tray or by using a set of mechanical probes that touch the feeder at a finite number of predetermined points. We show that this problem is equivalent to an NP-complete set-theoretic problem introduced as Discriminating Set, and present an O (n² m²) approximation algorithm to solve it with a ratio of 2 ln n. Furthermore, we prove that one can use an algorithm for Discriminating Set with ratio c log n to construct an algorithm for Set Covering with ratio c log n + O (log log n). Thus the ratio 2 ln n is asymptotically optimal unless NP DTIME (n^poly log n), a consequence of known results on approximating Set Covering. The complexity of subproblems of Discriminating Set is also analyzed, based on their relationship to a generalization of Independent Set called 3-Independent Set. Finally, simulation results suggest that sensing by point sampling is mostly applicable when poses are densely distributed in the plane.