Feed-rate optimization with jerk constraints for generating minimum-time trajectories

Abstract

Competitive pressure requires manufacturers to simultaneously address increasingly stringent constraints on both productivity and quality. From the perspective of numerically controlled (NC) machine tools, this means higher machining performance in terms of speed and accuracy. Conventional approaches to programming NC operations involve selecting a constant feedrate for a given operation to produce acceptable performance (operation time and contouring accuracy). In this paper, we examine the possibility of scheduling or varying the feedrate by taking into consideration the geometry of the contour that the machine is expected to follow and the physical capabilities of the machine (i.e., its maximum velocity, acceleration and jerk constraints). Previous work by the authors has addressed the efficient, off-line computation of time-optimal trajectories with constraints on velocity and acceleration. This paper introduces additional constraints on the permissible jerk (rate of change of acceleration) on the machine's axis. From a practical perspective, excessive jerk leads to excitation of vibrations in components in the machine assembly, accelerated wear in the transmission and bearing elements, noisy operations and large contouring errors at discontinuities (such as corners) in the machining path. The introduction of jerk into the feedrate scheduling problem makes generating computationally efficient solutions while simultaneously guaranteeing optimality a challenging problem. This paper approaches this problem as an extension of our previous bi-directional scan algorithm [23, 29]. A new acceleration-continuation procedure is added to the feedrate optimization algorithm to address jerk constraints and remove discontinuities in the acceleration profile. The algorithm maintains computational efficiency and supports the incorporation of a variety of state-dependent (such as position, velocity, acceleration and jerk) constraints. By carefully organizing the local search and acceleration continuity enforcing steps, a globally optimal solution is achieved. Singularities, or critical points, and critical curves on the trajectory, which are difficult to deal within optimal control approaches, are treated in a natural way in this algorithm. Several application examples and tests are performed to verify the effectiveness of this approach for high-speed contouring.