Subpixel Edge Localization via Ruled-Sigmoid Surfaces and Its Application for Precision Analysis of Cycloidal Gear Profiles

To overcome the limitations of single-dimensional data and low efficiency in traditional cycloidal gear inspection, a comprehensive machine vision-based method was proposed. A high-precision vision platform was established, and a Sigmoid surface-based edge detection algorithm was employed for sub-pixel edge localization. Logarithmic transformation combined with light intensity compensation was applied to correct saturation-induced errors. The pixel equivalent and compensation coefficient were systematically calibrated using a dot-matrix plate and gauge blocks. A sub-pixel tooth profile model in the physical coordinate system was reconstructed through pixel equivalent calibration, dynamic light intensity compensation, and multi-coordinate transformation. Comparative tests against a coordinate measuring machine (CMM) verified that the point-to-point deviation between the two measurement systems was within 10 μm (maximum 11.62 μm). The inherent tooth profile deviation of the tested cycloidal gears, which reflects the machining quality of workpieces, ranged from 24 μm to 37 μm. Multiple repeated tests prove that the system achieves a repeat positioning accuracy of 0.8 μm. Based on the measurement characteristics, a hybrid analytical method integrating Cartesian and polar coordinate systems was developed, enabling the simultaneous evaluation of critical geometric tolerances, such as the diameters of the center hole and crankshaft hole. The full inspection cycle for cycloidal gears was reduced to 13 s, which demonstrates a substantial efficiency improvement over traditional methods.