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To date, untethered magnetic devices actuated with a single rotating permanent magnet, such as active capsule endoscopes and magnetic microrobots, have been constrained to operate in positions where the rotating dipole field behavior is simple and easy to visualize. In this paper, we show how to generate a rotating magnetic field with any desired rotation axis, for magnetic device actuation, at any device position in space using a single rotating-magnet actuator. The methods presented can control untethered rotating magnetic devices, while the rotating actuator magnet follows trajectories independent of the untethered devices themselves. We demonstrate our methods by actuating rotating magnetic devices in a lumen. Applications include minimally invasive medical tasks requiring an untethered magnetic device to operate in natural lumen pathways of the body (e.g., the gastrointestinal system, the subarachnoid space of the nervous system, or vasculature).