Constrained Nonlinear Programming for Image-Based Hysteresis Compensation in Atomic Force Microscope Piezo Actuators

Compensation of Atomic Force Microscopy (AFM) image distortion due to the hysteresis of piezoelectric scanners is a critical task of AFM control systems. The main problem is to linearize the scans, i.e. transform them to the ones evenly spaced in the AFM images' XY lines. We propose a new approach to this task based on constrained optimization of transforms. The optimal transforms map original scans to nearly linear ones. These transforms (different for X and Y scans) can be used to map the distorted scans spaced evenly in time (but not in space) to the corrected (nearly even in space) XY image lines. This approach requires significant computing power which was a serious restriction for early AFM controllers but is eliminated now. Another, closer to traditional, approach is to use inverses of the optimal transforms in imbedded, usually implemented at FPGA, algorithms to transform linear-in-time voltages and apply them to XY scanners to produce scans that are evenly spaced in both time and space. In both approaches, constrained optimization provides rigorous design improvements over previously employed empirical methods. It also provides a rigorous approach to interpolate scaling that is different for X and Y as well as for varying scan sizes. These optimal transforms, e.g., selected from a set of polynomials, can also significantly reduce calculations required by Preisach model of hysteresis.