Angular interpolations for planar implicit curves

In multi-axis CNC machining, a cutting tool combines translation movements and rotational movements with respect to a workpiece. This entails CNC interpolation to take angular feedrates defining the speed of the rotational movement into account, while current interpolations are overwhelmingly based on linear federates defining the speed of the translation movement. The paper presents an approach using angular feedrate interpolations for planar tool paths in multi-axis machining, where a planar tool path is identified by a position curve assumed to be a planar implicit curve and a tool-orientation curve is obtained based on the position curve. The angular feedrate interpolations calculate the tool position and orientation at each sampling cycle based on specified angular feedrates and given tool paths. The paper relates analytically geometric properties of the prescribed path to two-dimensional rigid body motions along the path by angular feedrates and angular feed acceleration. The results are then used to develop the angular feedrate interpolation and improve the interpolation from the deviation analysis. The angular feedrate is further related to the linear feedrate and the contour error. Comparison between existing linear interpolation and proposed angular interpolation is also presented under the maximum allowed contour error constraint. Further simulations are carried out to illustrate the effectiveness of the interpolation scheme for complex curves.