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A magnetic manipulator for noncontact steering of magnetic objects is considered. This system utilizes a flexible array of permanent magnets, each equipped with a servomotor to independently control its direction. The total magnetic field produced by the magnets can be shaped effectively by adjusting the directions of all magnets, which in turn, provides an effective control over the magnetic force it applies to a magnetic object. The dynamics of this object under such controlled magnetic force is inherently unstable and is represented by a set of highly nonlinear state-space equations. Despite the nonlinear nature of these equations, it is shown that an optimally designed linear state feedback can successfully stabilize the object and steer it along arbitrary reference trajectories inside a reasonably large operation region. The key to this success is the optimal scheme of this paper for linearizing the dynamics of the magnetic object. 

Magnetic fields render a unique ability to control magnetic objects without a direct mechanical contact. To exploit this potential for a broad range of medical, microrobotics, and microfluidics applications, noncontact magnetic manipulators have been designed using both electromagnets and permanent magnets. By feedback control of these manipulators, magnetic objects can be precisely driven in the directions required by an application of interest. The feedback design process for these manipulators is normally complicated by their highly nonlinear nature, particularly for those utilizing permanent magnets. Yet, feedback linearization techniques can be applied to compensate for the nonlinear nature of most magnetic manipulators. This goal can be achieved by solving an underdetermined system of nonlinear algebraic equations. This paper adopts a homotopy continuation approach to solve this system of equations. It is shown by simulations that the proposed feedback linearization scheme drastically improves the control performance compared to the alternative control design methods used in prior work.