Summary: In the problem of parts feeding we give a class of feasible operations for reorienting a part, and we ask to find a fixed sequence of these operations which guarantee to bring the part into a known goal orientation from any possible initial orientation. Goldberg addressed this problem 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 that \(O(n)\) steps are sufficient.
In this paper, we prove Goldberg’s conjecture by explicitly constructing plans of at most \(2n- 1\) 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 of distinguishing among a finite set of parts using minimal sensing. Specifically, we assume that we are given a set \({\mathcal P}= \{P_1, P_2,\dots, P_k\}\) 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 \(P_i\in {\mathcal P}\), determines the index of this part.