The complexity of oblivious plans for orienting and distinguishing polygonal parts

In the problem ofparts feeding we are given a class of feasible operations for reorienting a part, and we are asked to find a fixed sequence of these operations which is guaranteed to bring the part into a known goal orientation from any possible initial orientation. Goldberg addressed this problem in [2], and showed that, for planar polygonal parts, there is always a sequence of simple operations which can be performed by a simple parallel-jaw gripper, which is guaranteed to orient the part (up to symmetry) without the use of any sensor information; he also conjectured thatO(n) steps are sufficient.In this paper we prove Goldberg's conjecture by explicitly constructing plans of at most2n − 1 steps for orienting polygonal parts in this model. We also give a lower bound on the number of steps required for such plans to show that this upper bound is tight.Finally, we extend these results to the problem ofdistinguishing among a finite set of parts using minimal sensing. Specifically, we assume that we are given a set of known polygonal parts, and a parallel-jaw gripper able to sense the distance between its jaws upon closure. We construct a simple oblivious plan of linear complexity which, when presented with a polygonal part, determines the index of this part.