Geometric Eccentricity and the Complexity of Manipulation Plans

Abstract. Complexity bounds for algorithms for robotic motion and manipulation can be misleading when they are constructed with pathological ``worst-case'' scenarios that rarely appear in practice. Complexity can in some cases be reduced by characterizing nonpathological objects in terms of intuitive geometric properties. In this paper we consider the number of push and push—squeeze gripper actions needed to orient a planar part without sensors and improve on the upper bound of O(n) for polygonal parts given by Chen and Ierardi in [1]. We define the geometric eccentricity of a planar part based on the length-to-width ratio of a distinguished type of bounding box. We show that any part with a given eccentricity can be oriented with a plan whose maximum length depends only on the eccentricity and not on the description complexity of the part. The analysis also applies to curved parts, providing the first complexity bound for nonpolygonal parts. Our results also yield new bounds on part feeders that use fences and conveyor belts.