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Permanent magnet manipulators provide a unique potential for safe operation of magnetically driven medical tools inside the human body for noninvasive surgical, imaging, and drug targeting procedures. These systems manipulate magnetic objects from a distance without direct contact, by generating a magnetic field using strong permanent magnets, and controlling the shape of this field properly. Control over the magnetic field is gained by displacement of the magnets using independent mechanical actuators for each magnet. However, interactions between the magnets result in a coupling between the actuators, which prevents them from independent and precise operation. This paper develops a multivariate, nonlinear feedback control to cancel the magnetic coupling between the actuators in effect. This feedback control incorporates a complex mathematical model of the magnetic interactions between the actuators, which does not admit a simple analytical form. Instead, this model is constructed numerically using the finite element method. The decoupling performance of the proposed feedback control is verified by numerical simulations.