An Analytical Transition Algorithm for Real-time CNC Machining of Linear Tool Path

This paper presents a locally optimal transition approach for computing a curve to smooth a corner defined by linear tool path, which generates curvature-continuous tool path in real time. Taking the locality of the optimization problem into account, transition unit including four data points of linear tool path is defined. With the aid of transition unit, the locally optimal model based on the sum of curvature extremes of two transition curves is constructed. On this basis, the objective of optimization is reformulated as a quadratic equation with respect to transition length under constraints of the lengths of linear segments and the threshold of approximation error. Unlike conventional numerical optimization problems, the locally optimal joint-point of two transition curves can be analytically determined with the help of the quadratic equation. And then the transition curves are easily obtained. To illustrate the effectiveness of the proposed approach, simulations on two linear tool paths are carried out respectively. The simulation results show that the feedrate obtained by the proposed approach is higher than that obtained by conventional methods. Hence, the machining efficiency is improved.