Minimum Bounds on the Number of Electromagnets Required for Remote Magnetic Manipulation

Numerous magnetic-manipulation systems have been developed to control objects in relatively large workspaces. These systems vary in their number of electromagnets, their configuration, and their limitations. To date, no attempt has been made to rigorously quantify how many electromagnets are required to perform a given magnetic manipulation task. For some tasks, such as controlling the field at a point, the answer is clear: the same number as dimension of control. For tasks that apply magnetic forces on an object, the answer is less clear, and some systems, which have more control magnets than kinematic degrees of freedom (DOFs), have demonstrated unexpected singularities that only arise at specific object orientations. This paper provides a general analysis for static electromagnetic systems rooted in the governing magnetic equations and proves an unintuitive result. That is, if only magnetic fields and forces are used to control an unconstrained magnetic object, four magnetic sources are required for 3-DOF force control and eight magnetic sources are required for orientation-independent 5-DOF force and heading control.