# Dwell time algorithm in deterministic polishing of a free-form surface based on the continuous tool influence function

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## Abstract

In deterministic polishing, solving the dwell time is one of the key factors. Usually, dwell time is solved by the tool influence function (TIF) and residual error. In previous research, single-point TIF (sTIF) is usually used in calculation, but it is not consistent with the TIF in the actual polishing process. In addition, when using the linear equation to solve the dwell time, a large TIF matrix results in a normal computer not having enough memory for calculation. In order to solve the above problems, a continuous TIF (cTIF) that changes with the polishing path is proposed: first, by the discretization method to simulate the continuous movement of the polishing tool in actual polishing, and then an optimized grouped least squares (LSQR) orthogonal decomposition algorithm is proposed to solve the dwell time. In this paper, an $x - y$ polynomial free-form surface with different initial residual errors (${\rm{RMS}} = {{30}}\;{\rm{nm}}$, ${\rm{PV}} = {{120}}\;{\rm{nm}}$; ${\rm{RMS}} = {{70}}\;{\rm{nm}}$, ${\rm{PV}} = {{280}}\;{\rm{nm}}$; and ${\rm{RMS}} = {{100}}\;{\rm{nm}}$ ${\rm{PV}} = {{400}}\;{\rm{nm}}$) were simulated by the proposed algorithm, respectively. The final residual error was ${\rm{RMS}} = {1.8}\;{\rm{nm}}$, ${\rm{PV}} = {13.3}\;{\rm{nm}}$; ${\rm{RMS}} = {2.6}\;{\rm{nm}}$, ${\rm{PV}} = {10.1}\;{\rm{nm}}$; and ${\rm{RMS}} = {2.8}\;{\rm{nm}}$, ${\rm{PV}} = {17.4}\;{\rm{nm}}$, respectively. The convergence rate of RMS and PV basically reached 95%, and the validity of the algorithm is proved.

© 2021 Optical Society of America

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