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This paper addresses the compensation of the dynamics-coupling effect in piezoscanners used for positioning in atomic force microscopes (AFMs). Piezoscanners are used to position the AFM probe, relative to the sample, both parallel to the sample surface (x and y axes) and perpendicular to the sample surface (z axis). In this paper, we show that dynamics-coupling from the scan axes (x and y axes) to the perpendicular z axis can generate significant positioning errors during high-speed AFM operation, i.e., when the sample is scanned at high speed. We use an inversion-based iterative control approach to compensate for this dynamics-coupling effect. Convergence of the iterative approach is investigated and experimental results show that the dynamics-coupling-caused error can be reduced, close to the noise level, using the proposed approach. Thus, the main contribution of this paper is the development of an approach to substantially reduce the dynamics-coupling-caused error and thereby, to enable high-speed AFM operation.