Tracking a magnetic nanoparticle in 3-D with a magnetic force microscope

In this paper we introduce a scheme for tracking a magnetic nanoparticle in 3-D with the tip of a magnetic force microscope. The stray magnetic field of the magnetic particle induces a shift in the phase of oscillation of the tip of the MFM. We present a feedback control law which translates the measurement of this phase shift into actuator commands to the MFM and prove that the trajectory of the tip converges to a neighborhood of the magnetic particle. This geometric control law depends only on the derivative of the potential along the trajectory of the tip and in particular does not rely on any detailed prior knowledge about the nanoparticle. The results of simulation studies are shown to illustrate the algorithm.