Smooth and Time-Optimal Trajectory Planning for Multi-Axis Machine Tools

Abstract

This thesis presents novel methods for feedrate optimization and toolpath smoothing in CNC machining. Descriptions of the algorithms, simulation test cases, and experimental results are presented. Both feedrate optimization and toolpath smoothing are essential for increasing manufacturing efficiency while retaining part quality in CNC machining. The application of high-speed machining also necessitates the use of high feedrates, and smooth toolpaths which can be safely traversed at high feeds. However, problems occur when the feedrate is increased without check. High tracking error in machining may cause part tolerance errors. Transient vibrations due to jerky movement can lead to poor part surface quality. High speed trajectories may also demand greater torque than what the feed drives are capable of producing, which affects the motion controller’s ability to follow the trajectory correctly. The condition of the machine is also a concern, with the potential for damage or excessive wear on the machine’s components, if excessive axis velocity or jerk (i.e., rate of change of acceleration) is commanded. The feedrate scheduling algorithm developed in this thesis combines linear and nonlinear programming in a dual-windowed implementation. Linear programming (which is computationally fast) is used to quickly provide a near-optimal guess, based on axis velocity, acceleration, and jerk constraints. The solution is then refined through the use of nonlinear optimization. In the latter step, requiring more computations, the commanded motor torque and expected servo error are constrained directly, leading to shorter movement time. A windowing alignment procedure is presented which allows for these two optimization methods, each with different problem constraints and solutions horizons, to work in tandem with one another without risking infeasible boundary conditions between the windows. The algorithm is validated in simulation and experiment studies. Case studies analyzing the parameters of the optimization algorithm are also presented, and the configuration which is most computationally efficient is determined. A toolpath generation method is presented in which Euler-spiral pairs are used to smooth sharp corners, with an algorithm that integrates directly with the developed feedrate optimization The result is an exactly arc-length parametrized, G2-continuous toolpath whose axis derivatives can be computed very efficiently, which helps reduce the overall computation time. A repositioning toolpath method is also developed to reduce the cycle time of multi-layer contouring operations. This method replaces circular arc based repositioning segments between contouring passes (commonly used in industry) with a smooth Euler spiral based curve. This avoids tangent and curvature discontinuities, allowing for smoother motion with lower velocity and acceleration demands, while also reducing the overall motion. The repositioning toolpath has also been integrated with feedrate optimization and validated in simulation results.