Kinematic Modeling of a Small-Tooth-Difference Planetary Involute Reducer

Abstract

This paper develops a general matrix-based kinematic model of a small-tooth-difference planetary involute reducer operating at the minimum tooth-number difference. The mechanism comprises a fixed internally toothed central gear, two identical externally toothed satellites mounted on oppositely phased eccentric supports of the input shaft, an intermediate ball coupling between the satellites, and a second ball coupling transmitting motion to the coaxial output shaft. The formulation combines generalized coordinates, an angular constraint matrix, a transmission vector, and block-matrix representations of satellite orbital and rigid-body motion. The satellites undergo plane-parallel motion combining circular orbital translation with slow self-rotation, the latter constituting the useful reduced motion transmitted to the output. The ball couplings are treated only through geometric compatibility conditions. A parametric study for several tooth-number differences evaluates transmission-ratio sensitivity, eccentricity variation, reconfiguration using interchangeable eccentric bushings, and numerical consistency of the proposed matrix model over one input revolution under ideal rigid-body assumptions.