Non-vector space approach for nanoscale motion control

As the advancement of nanotechnology, it is possible to manipulate structures at nanoscale with various nanomanipulation tools such as scanning probe microscopes. To achieve successful manipulations, precise motion control is required, especially for objects with sizes from subnanometer to several nanometers. To address this issue, this paper presents an image based non-vector space control approach. Considering images obtained from the microscopes as sets, the dynamics of the system can be formulated in the space of sets. Since the linear structure of the vector space is not available in this space, this method is called the non-vector space control. With the dynamics in the non-vector space, we formulate the stabilization problem and design the controller. The stabilization controller is tested with images obtained by atomic force microscopes, and the results verify the proposed theory. The method presented in this paper does not rely on external sensors for position feedback. Moreover, unlike the traditional image based control method, we do not need to extract features from images and track them during the control process. Finally, the control precision can be as good as the imaging resolution. The approach presented in this paper can also be extended to other systems where the states can be represented as sets.