A twist exponential approach to gear generation with general spatial motions

This paper presents a novel formulation of gear generation with general spatial motions obtained by parameterizing the computer numerically controlled (CNC) machine employed for cutting via the product-of-exponentials (POE) formula. By exploiting its interpretation as a serial kinematic chain, a systematic methodology is presented for efficiently computing the forward kinematics of the generating tool relative to the gear blank, which is essential, e.g., in the derivation of the envelope (tooth) surface. A key feature of the proposed method is that elementary motions allowed by the machine joints are parameterized directly via twist exponentials, with the twofold advantage of: (1) avoiding the introduction of a long chain of reference frames; (2) obtaining the rigid-body velocity (twist) of the enveloping motion directly from the parametrization. A layout typical of the face-milling process for hypoid gears is chosen as a paradigm in the unfolding of the theory, although the approach can be adapted to any process and arrangement. As examples of application, explicit expressions pertinent to gear generation between two fixed axes, and for a set-up typical of a 9-axis Gleason CNC universal motion concept (UMC) machine are presented.