A three-dimensional tracking algorithm for efficient construction of the feasible space of tool axis for a conical toroidal-end cutter in five-axis machining

Conical toroidal-end cutters are being increasingly adopted for multi-axis milling of free-form surfaces in industrial manufacturing, benefiting from their higher cutting speed and stiffness than the conventional cylindrical ball-end cutters with the same radius. Calculating the feasible space of tool axis (FSTA) for such cutters in complex environments containing free-form surface obstacles, for any given cutting contact position with its associated normal vector, is an extremely time-consuming task. The computational challenge arises because the conventional brute-force approach needs to check collision for a huge number of sampled tool axes. To address this challenge, this paper developed the state-of-the-art boundary-focused computational framework for constructing FSTA, which is featured by direct identification of characteristic points corresponding to the critical tool axes located on the check surfaces, applicable to conical toroidal-end cutters. The essential breakthrough is the derivation of geometric properties of the characteristic points for toroidal-end cutter whose head center is non-fixed and varies with tool axis. Based on these theoretical insights, a tracking-based numerical algorithm for efficiently constructing FSTA is then described. Simulation tests validate that our algorithm significantly enhances the computational efficiency while simultaneously improving the accuracy of FSTA boundary.