Three-Dimensional Contouring Control: A Task Polar Coordinate Frame Approach

In order to characterize the surface quality of machined parts, researchers have proposed a family of coordinate transformation-based approaches to deal with the contour following problem, which alleviate the computational consumption of calculating the contouring error. However, there are some drawbacks in the existing approaches, e.g., the limitation in two-dimension and the specification of certain contour error estimation methods. In this paper, we propose a novel task polar coordinate frame based on the spatial snapshot at a given time. The advantages of the proposed frame possess the following highlights. First, the frame requires no specification for a certain contouring error estimation approach, which generalizes its applications to desired contours with different curvatures or torsions. Second, the frame can be applied in both two- and three-dimensions. Third, the dynamics of the three-dimensional contour following system can be equivalently transformed into a two-dimensional problem along the radial and angular axes in the proposed frame, and the obtained problem is thus further transformed into a two-dimensional decoupled regulation problem, which can be solved by the computed-torque control technique and the proportional-derivative controller. The experiments are performed on an industrial three-axis machine tool, and the results confirm the capability of the proposed frame in addressing the three-dimensional contour following problem, and its generalization to different contour error estimation approaches. Moreover, we examine the computational cost of our frame and prove that it can be applied in real-time industrial control systems.