Vector model-based workpiece update in multi-axis milling by moving surface of revolution

This paper presents a parametric approach to updating workpiece surfaces in a virtual environment. The workpiece surfaces are represented by a series of discrete vectors, which may be orientated in different directions. The methodology is developed for multi-axis machining in which a tool can be arbitrarily oriented in space. The cutter is modeled as a surface of revolution, which is a canal surface formed by sweeping a sphere with varying radius along a spine curve. To define the tool swept envelope, the cutter surfaces are decomposed into a set of characteristic circles which are generated by a two-parameter family of spheres. Then, the grazing points, at which the discrete vectors can intersect the tool envelope, are obtained by considering the relationships between these circles and feed vector of the cutter. From this, the envelope-vector intersections are transformed into a single-variable function. Examples of this technique are generated for typical milling tools with both linear and circular spine curves. The vector/tool envelope intersection calculations for cutters with linear spine curves can be performed analytically. However, the intersection calculations for cutters having circular spine curves require solving a system of nonlinear equations. For this purpose, a root-finding analysis is developed for guaranteeing the root(s) in the given interval. Finally, to improve the efficiency when updating the workpiece, a vector localization scheme is developed based on the Axis-aligned Bounding Box method.