Upper and Lower Bounds for Programmable VectorFields with Applications to MEMS and Vibratory

Programmable vector elds can be used to control a variety of exible planar parts feeders. These devices can exploit exotic actuation technologies such as ar-rayed, massively-parallel microfabricated motion pixels or transversely vibrating (macroscopic) plates. These new automation designs promise great exibility, speed, and dexterity|we believe they may be employed to orient , singulate, sort, feed, and assemble parts. However, since they have only recently been invented, programming and controlling them for manipulation tasks is challenging. When a part is placed on our devices, the programmed vector eld induces a force and moment upon it. Over time, the part may come to rest in a dynamic equilibrium state. By chaining together sequences of vector elds, the equilibrium states of a part in the eld may be cascaded to obtain a desired nal state. The resulting strategies require no sensing and enjoy eecient planning algorithms. This paper begins by describing our experimental devices. In particular, we describe our progress in building the M-Chip (Manipulation Chip), a massively parallel array of programmable micro-motion pixels. As proof of concept, we demonstrate a prototype M-Chip containing over 11,000 silicon actuators in one square inch. Both the M-Chip, as well as macroscopic devices such as transversely vibrating plates, may be programmed with vector elds, and their behavior predicted and controlled using our equilibrium analysis. We demonstrate lower bounds (i.e., impossibility results) on what the devices cannot do, and results on a classi-cation of control strategies yielding design criteria by which well-behaved manipulation strategies may be developed. We provide suucient conditions for program-mable elds to induce well-behaved equilibria on every part. We deene composition operators to build complex strategies from simple ones, and show the resulting elds are also well-behaved. We discuss whether elds outside this class can be useful and free of pathology. Using these tools, we describe new manipulation algorithms. In particular, we improve existing planning algorithms by a quadratic factor, and the plan-length by a linear factor. Using our new and improved strategies, we show how to simultaneously orient and pose any part, without sensing, from an arbitrary initial conng-uration. We relax earlier dynamic and mechanical assumptions to obtain more robust and exible strategies. Finally, we consider parts feeders that can only implement a very limited \vocabulary" of vector elds (as opposed to the pixel-wise programmability assumed above). We show how to plan and execute parts-posing and orienting strategies for these devices, but with …