Error Compensation for Optical Encoder Based on Variational Mode Decomposition With a Coarse-to-Fine Selection Scheme

Abstract

To further improve the compensation performance of the optical encoder at a reasonable speed, a novel error compensation method is proposed based on variational mode decomposition (VMD) with a coarse-to-fine selection scheme. We make the first attempt to introduce the VMD into error compensation for the optical encoder due to its excellent abilities of mode-mixing suppression and low computational cost. First, the VMD with an adaptive parameter selection scheme is proposed to decompose the measurement error into several band-limited intrinsic mode functions (BLIMFs). Then, a coarse-to-fine selection scheme is proposed to adaptively select eligible BLIMFs, which are used to reconstruct the systematic error from the measurement error. The coarse selection scheme is proposed based on permutation entropy (PE) to select the candidate BLIMFs with low randomness and complexity. Also, a contribution factor is defined to finely select the eligible BLIMFs from the candidate BLIMFs, which is defined by combining PE and optimal effective singular rank of each candidate BLIMF. Experimental results indicate that the proposed method is superior to the existing compensation methods, which can achieve an excellent compensation precision of root-mean-square error (RMSE) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.335 ~\mu \text{m}$ </tex-math></inline-formula> and 95% confidence interval (−0.015, 0.014) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> .