Classification and Lower Bounds for MEMS Arrays and Vibratory Parts Feeders: What Programmable Vector Fields Can (and Cannot) Do --Part I

Programmable vector fields can be used to control a variety of flexible planar parts feeders. These devices can exploit exotic actuation technologies such as arrayed, massively-parallel microfabricated motion pixels or transversely vibrating (macroscopic) plates. These new automation designs promise great flexibility, 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. By chaining together sequences of vector fields, the equilibrium states of a part in the field may be cascaded to obtain a desired final state. The resulting strategies require no sensing and enjoy efficient planning algorithms. This paper begins by describing our experimental devices. In particular, we describe our progress in building the {\sc M-Chip} (\underline{m}anipulation \underline{chip}), a massively parallel array of programmable micro-motion pixels. As proof of concept, we demonstrate a prototype {\sc M-Chip} containing over 11,000 silicon actuators in one square inch. Both the {\sc M-Chip}, as well as macroscopic devices such as transversely vibrating plates, may be programmed with vector fields, and their behavior predicted and controlled using our {\em equilibrium analysis}. We demonstrate lower bounds (i.e., impossibility results) on what the devices {\em cannot} do, and results on a classification of control strategies yielding design criteria by which well-behaved manipulation strategies may be developed. We provide sufficient conditions for programmable fields to induce well-behaved equilibria on every part placed on our devices. We define {\em composition operators} to build complex strategies from simple ones, and show the resulting fields are also well-behaved. We discuss whether fields outside this class can be useful and free of pathology.