Interpolating Control for Precision Positioning affected by pre-sliding Friction

For wafer inspection, an X-Y stage is used to position the wafer under a probe, such as X-ray or similar, with an accuracy of tens of nano-meters or less. The point-to-point movement typically consists of three phases: “move” when the position is controlled to follow a so-called S-curve, “settle” when the position is made to enter the convergence window, and “idle” when residual vibrations are damped and the position reaches the set-point with required accuracy. The “move” and “settle” plants are characterized by double-integrator dynamics with viscous and Coulomb friction, and the “idle” plant by pre-sliding friction dynamics that can be modeled as one or more mass-spring-damper systems. Taking into account high-frequency parasitic resonances and anti-resonances, non-linearities, and uncertainties, it is in general difficult, or impossible, to design one linear regulator for all three phases. Hence, one regulator is generally applied for “move” and “settle”, and another one for the ”idle” phase. The problem of switching between the regulators, preferably in a bump-less way in order not to evoke the parasitic resonances, has been treated in an ad-hoc manner in the literature and in industrial practice. Here, we suggest bump-less transfer between the regulators based on interpolating control, using a recent, simple friction model that includes pre-sliding.