Complete algorithms for feeding polyhedral parts using pivot grasps

To rapidly feed industrial parts on an assembly line, Carlisle et al. (1994) proposed a flexible part feeding system that drops parts on a flat conveyor belt, determines position and orientation of each part with a vision system, and then moves them into a desired orientation. When a part is grasped with two hard finger contacts and lifted, it pivots under gravity into a stable configuration. The authors refer to the sequence of picking up the part, allowing it to pivot, and replacing it on the table as a pivot grasp. The authors show that under idealized conditions, a robot arm with four degrees of freedom (DOF) can move (feed) parts arbitrarily in 6 DOF using pivot grasps. This paper considers the following planning problem: Given a polyhedral part shape, coefficient of friction, and a pair of stable configurations as input, find pairs of grasp points that will cause the part to pivot from one stable configuration to the other. For a part with n faces and m stable configurations, the authors give an O(m²n log n) algorithm to generate the m×m matrix of pivot grasps. When the part is star-shaped, this reduces to O(m²n). Since pivot grasps may not exist for some transitions, multiple steps may be needed. Alternatively, the authors consider the set of grasps where the part pivots to a configuration within a “capture region” around the stable configuration; when the part is released, it will tumble to the desired configuration. Both algorithms are complete in that they are guaranteed to find pivot grasps when they exist