The movement of a catheter through a medium, which may be living tissue such as a human brain, is controlled by mechanically pushing a flexible catheter having a magnetic tip through the medium and applying a magnetic field having a magnitude and a direction that guides the mechanically-pushed catheter tip stepwise along a desired path. The magnetic field is controlled in a Magnetic Stereotaxis System by a processor using an adaptation of a PID (proportional, integral, and derivative) feedback method. The magnetic fields are applied by superconducting coils, and the currents applied through the coils are selected to minimize a current metric.