Local minimum time trajectory planning for five-axis machining with or without deflection

This paper presents the algorithms to obtain the local minimum-time trajectory planning for the five-axis machining with or without the tool deflection. The tool and workpiece are first combined as a closed-chain system of rigid bodies. The tool path and shape are simplified to the tool-workpiece system curve and tool line. The forward and inverse kinematics are applied to obtain the kinematic relation between the system path and the position of the five motors. Based on the kinematic equations, the velocity, acceleration, and jerk of each motor can be derived. The motion trajectory of the end-effector of this system can be described through a sequence of intervals. In order to guarantee the dimension accuracy, the quintic polynomial was applied to fit the curve on each individual interval. The genetic algorithm (GA) with one generation is applied to find the minimum time period at every interval on the system curve in the case that the system has no deflection on the reference path. When there is the deflection on the system path, an algorithm is developed to find its projection at the i^th interval on the reference path without deflection through the potential field method, and then the local minimum time periods at two new intervals composed with deflection, its projection, and the (i+1)^th reference point, which is on the curve without deflection, respectively. The derived kinematic equations and the proposed algorithms are verified through the simulation.