Matrix Formulation of Rigid-Body and Contact Kinematics of Ball Couplings in a Small-Tooth-Difference Planetary Involute Reducer

Small-tooth-difference planetary involute reducers require an output coupling capable of transmitting the reduced rotation of eccentrically moving satellites to a coaxial output shaft. This study develops a matrix formulation of the rigid-body and contact kinematics of two successive three-ball couplings integrated into such a reducer. The method combines generalized-coordinate, constraint, transmission, rotation, block, and circumferential-distribution matrices with three-dimensional contact bases, skew-symmetric operators, projection matrices, and a continuous least-squares formulation along the finite curved ball–groove contact line. The model distinguishes absolute and relative ball-center motion and shows that the two coupling stages share the same relative-motion structure, differing only by a constant geometric scale factor. Perfect no-slip rolling cannot generally be satisfied simultaneously along the entire finite contact line for nonzero transmitted motion; a uniformly weighted minimum-sliding solution therefore provides a unique reference rigid-body ball rotation, together with rolling, spin, and residual-sliding distributions. The analysis also establishes precession of the instantaneous ball-rotation axes, distinguishes ball-center trajectory closure from ball-orientation closure, and quantifies the influence of normalized groove depth. The resulting framework provides a compact basis for subsequent contact-mechanics and tribological analyses.