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

We consider the tool orientation problem of a 5-axis CNC machine that cuts along a curved toolpath on a smooth workpiece surface. While the tool axis is required to maintain a fixed angle against the surface normal to achieve a constant cutting speed, it is allowed to rotate about the surface normal in an arbitrary manner. We control the variation of this azimuthal angle so that the instantaneous speed of the tool axis motion is minimal, which is a judicious choice for the efficient and robust machine performance. The ensuing curve traced by the tool axis turns out to be a generalized tractrix on the unit sphere determined by the curve traced by the surface normal vector, which is characterized by the property that the great circle determined by the corresponding points on the two curves is always tangent to the tractrix.