Interference-free NC machining using spatial planning and Minkowski operations

Abstract

NC machining involves a description of the erosion of a stock of material by the milling tool as well as interference avoidance between tool holder and the stock, and between the tool and the model of the result to be obtained. In spatial planning an object is placed or moved among one or more obstacles without interfering with them. The object is shrunk to a point o and the obstacles are dilated with the object. The dilated obstacles then represent the forbidden region as the complement of the free space in which o can be positioned. Hence, interference avoidance for NC machining involves both a tool holder spatial planning problem and a tool spatial planning problem. In the tool holder spatial planning problem the tool holder is the object and the stock is the single obstacle. In the tool spatial planning problem the tool is the object and the model of the result to be obtained is the single obstacle. In both problems the tool holder and the tool are shrunk to the same point o. Then the forbidden region for positioning o is the union of the dilated stock and the dilated model and the tool path is extracted from the boundary between the forbidden region and the free space. In the field of mathematical morphology the Minkowski addition and subtraction provide a formalism to describe dilation and erosion. In this paper we apply this formalism to state the NC machining problem, and to describe, implement and compare two important grid height machining methods. We illustrate these methods with some practical examples.