On the estimation of continuous mappings from cradle-style to 6-axis machines for face-milled hypoid gear generation

Modern Computer Numerically Controlled (CNC) machines for grinding face-milled hypoid and spiral bevel gears have six axes and are able to perform complex motions with almost free-form ease-off topography correction capabilities. However, in the gear literature and in the common practice machine-settings, corrections are usually performed within the classical cradle-style framework. We believe that, working in the 6-axis framework, may have advantages with respect to correction performances, and numerical conditioning of the resulting optimization problem. However, since for historical and practical reasons basic kinematic settings are usually given within the cradle-style framework, an imperative intermediate step is the translation of the cradle-style kinematics into the 6-axis (minimal) layout. Therefore, in this paper the estimation of the continuous mappings from the cradle-style to the 6-axis kinematics is investigated. In particular, the performances of two methods are assessed: (i) an existing central difference method (CDM) approach, and (ii) an original Least Squares Method (LSM) approach. Numerical results show that the LSM approach presents some advantages over the CDM approach in terms of accuracy of the reconstructed surface, numerical stability, and ease of implementation.