The use of concave and convex functions to optimize the feed-rate of numerically controlled machine tools

Optimality and computational efficiency are two very desirable but also competitive attributes of optimal feed planning. A well-designed algorithm can vastly increase machining productivity by reducing tool positioning time subject to limits of the machine tool. The nonlinear optimization problem aims to achieve the highest possible feed along the tool path, while limiting the speed of the actuator level, acceleration and Jerk profiles. Methods proposed in the literature either use rather complex nonlinear optimization solvers, such as Sequential Quadratic Programming, use iterative heuristics that extends computation time, or use conventional assumptions that reduce computation time but lead to slower tool motion.The problem of optimal feed-rate planning along a curved tool path for multi-axis CNC machines with a Jerk limit for each axis is addressed. However, the use of Jerk (rate of change of acceleration) into the feed-rate scheduling problem causes generating both, computationally efficient solutions and simultaneously guaranteeing optimality, is a challenging problem. To solve this problem, we propose the approach of modifying a Jerk parameter, through the use of a pair of convex and concave functions, including aggregation functions. In this technique, the suggested algorithm indicates the points at which the increasing convex/concave function and the adequate dual decreasing function modeling the Jerk parameter is used.