Feasible region determination without local interference of 3 + 2 axis machining for arbitrary revolving tools

The 3 + 2 axis machining technology is of high feasibility and strong rigidity. Its feasible region determination is a prerequisite and indispensable. This paper focuses on feasible region determination without local interference for arbitrary revolving tools in 3 + 2 axis machining. A unified definition of revolving tools is firstly raised, and the normal vector surfaces are mapped to areas on a same unit sphere. The feasible region determination is transformed into a covering problem between two areas on a unit sphere, which are the normal vector surface of the tool and the negative normal vector surface of the machining area. This method is independent of tool geometry types. If the normal vector surfaces are determined, then the feasible region is obtained. To get more maneuverability, in the actual determination process, the negative normal vector surface of the machining area is approximated by a revolving surface on a unit sphere which is called as its γ-revolving surface in this paper, and the error is analyzed. The γ-revolving surface is obtained by the steepest descent method to search the minimum revolving surface in a certain sense. The covering problem between two revolving surfaces on a unit sphere is solved on a plane, which equals to consider the covering between their generatrices. In the validation, the feasibility of tool axes in feasible regions’ different positions is verified, and the method is used in tool selection.