Time-optimal feedrate planning for real-time CNC machining via transformation between parametric and time spaces

This paper presents a new jerk-limited feedrate optimization algorithm for spline toolpaths with a regularized time-scaled nonlinear programming formulation, to increase the productivity of multi-axis machines with serial kinematics, such as machine tools or industrial robots. By combining the quadratic programming based solution with a pseudo-jerk estimate as its warm start, kinematic limits in both joint and Cartesian space can be handled directly in a time-discretized form without the need for arc length parameterization of toolpaths. The original time-optimal formulation is extended to a regularized form using integral smoothing to improve the smoothness of the feedrate profile. The associated optimization problem is formulated in a sparse form by direct transcription and can be solved efficiently and reliably using the interior point method. In addition, designed for its use in industrial CNC applications, a sequential windowing strategy is introduced to deal with long toolpaths with a large number of constraint checkpoints. This strategy allows its potential integration with the look-ahead functionality in a real-time capable CNC kernel. The constraint feasibility and the computational efficiency of the proposed algorithm are validated on a 6 DOF KUKA industrial robot kinematics with butterfly-shaped NURBS toolpaths. The computation time is kept well below the actual finishing time (about 10% measured on an i9 CPU with single core execution), and shows an almost linear dependence on the number of constraint checkpoints. The planning performance and computational robustness have also been validated experimentally on a milling machine following a 3D flower-shaped freeform toolpath with 10 000 constraint checkpoints. Compared to an industrial numerical control kernel using an S-curve profile-based planning approach, the proposed algorithm reduces both the finishing time and the mean Cartesian contour error in the 3D tracking experiment by 17% and 21% respectively.