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It has been shown that when a magnetic dipole, such as a permanent magnet, is rotated around a fixed axis such that the dipole is perpendicular to the axis of rotation, the magnetic field vector at every point in space also rotates around a fixed axis. In this paper, we reformulate this phenomenon using linear algebraic techniques, which enables us to find the necessary dipole rotation axis to make the magnetic field at any desired point in space rotate about any desired axis. To date, untethered magnetically actuated tools (e.g., capsule endoscopes, rolling spheres, and helical-propeller microswimmers) controlled with a single rotating permanent magnet have been constrained to operate in positions where the rotating field behavior is simple and easy to visualize. We experimentally demonstrate that the results of this paper can be used to control a variety of untethered, rotating magnetic devices in any position even while the rotating permanent magnet follows trajectories independent of the devices themselves. This method constitutes a substantial step toward making a great deal of prior laboratory research regarding rotating magnetic microrobots and capsule endoscopes clinically feasible.

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