Tractrix-based tool orientation control for 5-axis CNC machining

We consider a 5-axis CNC machine whose tool axis maintains a fixed angle with respect to the surface normal as the machine tool cuts a given path on a smooth surface. The tool axis a(ξ) lies on a cone of constant angle ψ about the surface normal n(ξ) at each point of the path, but its azimuthal position on this cone remains indeterminate. Geometrically speaking, given a curve n(ξ) on the unit sphere, we have to choose a(ξ) among such curves on the sphere that satisfy a(ξ) · n(ξ) = cosψ. We discuss several schemes to resolve the indeterminacy by focusing on the minimization of the residual motion of tool axis. The first method is to force the tangential component of a(ξ) to the surface to be parallel along the path, in other words requiring the component to be rotation-minimizing with respect to n(ξ). The second method is to find a(ξ) on the unit sphere with minimal length under the aforementioned constraint, which is formulated in terms of variational calculus. Finally we introduce a tractrix-like curve on the sphere that orient itself in such a way that locally minimizes its length. Although this curve does not achieve globally minimal length, its simple derivation and length-minimizing property can be more appealing in practice. Chang Yong Han, Kyung Hee University.