A geometric approach to designing a programmable force field with a unique stable equilibrium for parts in the plane

In automated assembly, before parts can be put together, they often have to be appropriately oriented and positioned. The device performing this task is generally referred to as a part feeder. A new class of devices for non-prehensile distributed manipulation, such as MEMS actuator arrays, vibrating plates, etc., provide an alternative to traditional mechanical platforms for part feeding. These devices can be abstracted as programmable vector fields. Manipulation plans for these devices can therefore be considered as strategies for applying a sequence of fields to bring parts to some desired configurations. Typically, to uniquely orient and position a part, several fields have to be sequentially employed. Previously, it has been proven that there exists a combination of the unit radial field and a constant field that induces a unique stable equilibrium for almost any part. However, that work focuses mainly on an existential proof and fails to address how to compute the field for a given part. We propose a radically different field with a proof confirming that the field induces a unique stable equilibrium for almost any part. This proof leads us to a method for computing a single field for orienting a given part, together with the corresponding stable equilibrium configuration of the part.