On fence design and the complexity of push plans for orienting parts

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of a polygonal part by a sequence of push actions. We show that any polygonaJ part can be oriented by a sequence of fences placed along a conveyor belt, thereby settling a conjecture by Wiegley et al. [25], and present the first polynomial-time algorithm to compute the shortest such sequence. The algorithm is simple and runs in time 0(n3 log n), where n is the number of vertices of the part. Even though pathological parts can be constructed that require Q(n) push actions, it turns out that almost all parts can be oriented using a small number of pushes. We deduce a new bound on the length of the shortest push plan that depends on the thinness – or eccentricity – of the part. The bound shows that only O(1) pushes are required for the large class of parts with non-square minimum-width bounding boxes.