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.

► The POE parameterization of the cradle-style (CS) and the 6-axis (6M) minimal hypoid generators is presented.
► The pointwise kinematic inversion between the CS machine and the 6M generator is fully detailed.
► A comparison between an existing method and an original method for defining continuous 6M joint functions is presented.
► The new method leads to better reconstruction of the CS kinematics and to reduced errors in the ground pinion surface.