Point Pair-Based Expression of Cutter Swept Envelopes in Five-Axis Milling

Abstract

This paper proposes an efficient method of formulating cutter swept envelopes in five-axis milling based on conformal geometric algebra. The surface of the rotary cutter is represented as a canal surface, as the envelope of a one-parameter family of spheres. The swept envelopes generated by the general rigid motion of the cutter can be regarded as the envelopes of a two-parameter family of spheres. With the help of the dual representations of spheres and planes in conformal space, a point pair-based expression of the swept envelopes is formulated analytically, which represents two surfaces uniformly. By using the present method, the swept surfaces of a conical cutter, a drum-shaped cutter, and a toroidal cutter are illustrated. As an application, two types of collision detection methods for 5-axis milling are presented based on the point pair-based envelope expression. The benchmark examples demonstrate that the use of the proposed envelope expression significantly improves computational efficiency.