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Optimal inversion of system dynamics can be used to design inputs that achieve precision output tracking. However, a challenge in implementing the optimal-inversion approach is that the resulting inverse input tends to be noncausal. The noncausality of the optimal inverse implies that the desired output trajectory must be pre-specified and cannot be changed online. Therefore, the optimal inverse can only be used in trajectory-planning applications (where the desired output is known in advance for all future time). The main contribution of this article is the development of a technique to compute the noncausal optimal inverse when the desired output trajectory is known in advance for only a finite time interval. This future time interval, during which the desired output trajectory is specified, is referred to as the preview time. Additionally, this article develops a time-domain implementation of the optimal inverse and quantifies the required preview time in terms of the specified accuracy in output tracking, the system dynamics, and the cost function used to develop the optimal inverse. The proposed approach is applied to precision (subnanoscale) positioning of a scanning tunneling microscope (STM), which is a key enabling tool in emerging nanotechnologies. Experimental results are presented which show that finite preview of the desired output trajectory is sufficient to operate the STM at high speeds.