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Positioning applications are very important in a variety of industrial processes, including automatic assembly. This paper proposes a technique for positioning applications that involves employing a platform subjected to planar oscillations along circular, elliptical, and complex trajectories. Dynamic and mathematical models of the motion of a part on the platform were developed to investigate the motion characteristics of the part. The research showed that when the platform was excited in two perpendicular directions by sinusoidal waves, different trajectories of the part’s motion could be obtained by controlling excitation parameters such as the frequencies and amplitudes of the waves and the phase shift between the waves. Furthermore, by adjusting these parameters, the average displacement velocity of the part could be controlled. The results demonstrate that the part can be moved in any direction at a given velocity and can be subjected to complex dense positioning trajectories. Therefore, such a platform can be applied in feeding, positioning, and manipulation tasks.