Radial Spectral Projection-Based Axial Localization Using an Encoded Pattern for High-Resolution Four-DOF Motion Measurement

High-precision, multi degree-of-freedom (DOF) motion sensing across scales is critical for advanced mechatronic systems. Here, we propose a pseudoperiodic encoded pattern-based visual sensing method for four-DOF (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</i>, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ_Z</i>) pose estimation. Lateral displacement and in-plane rotation are extracted <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">via</i> phase and spectral analysis, while axial localization is achieved by exploiting defocus-induced spectral variations combined with a template-matching strategy. Specifically, the two-dimensional (2D) frequency spectrum is converted into a 1D radial vector, which is then matched against a pre-calibrated, depth-dependent model to estimate the axial position. By applying principal component analysis, lateral-axial coupling is effectively suppressed. In addition, defocus-phase correction and a point spread function model associated with axial positions are incorporated for real-time image restoration, enabling robust lateral positioning under defocused imaging conditions. The proposed method supports decimeter-scale in-plane displacement measurement and in-plane rotation estimation, together with nanometer-level axial localization within a validated range of ±20 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</i>m. Experimental results demonstrate root-mean-square resolutions of 0.57 nm, 0.61 nm, 1.18 nm, and 2.45 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</i>rad along the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</i>, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ_Z</i> axes, respectively. This approach provides an effective solution for ultrahigh-precision pose sensing and motion control across multiple scales.