Estimation of the shear force in transverse dynamic force microscopy using a sliding mode observer

In this paper, the problem of estimating the shear force affecting the tip of the cantilever in a Transverse Dynamic Force Microscope (TDFM) using a real-time implementable sliding mode observer is addressed. The behaviour of a vertically oriented oscillated cantilever, in close proximity to a specimen surface, facilitates the imaging of the specimen at nano-metre scale. Distance changes between the cantilever tip and the specimen can be inferred from the oscillation amplitudes, but also from the shear force acting at the tip. Thus, the problem of accurately estimating the shear force is of significance when specimen images and mechanical properties need to be obtained at submolecular precision. A low order dynamic model of the cantilever is derived using the method of lines, for the purpose of estimating the shear force. Based on this model, an estimator using sliding mode techniques is presented to reconstruct the unknown shear force, from only tip position measurements and knowledge of the excitation signal applied to the top of the cantilever. Comparisons to methods assuming a quasi-static harmonic balance are made.