Conformal slit mapping based spiral tool trajectory planning for ball-end milling on complex freeform surfaces

We propose a spiral-based complete coverage strategy for ball-end milling on perforated freeform surfaces, using conformal slit mapping to produce compact, smooth, and evenly spaced toolpaths. Extending prior methods, our algorithm embeds continuous spiral trajectories directly into 3D perforated surfaces without requiring cellular decomposition or artificial boundaries. Key challenges include extending slit mapping to discrete mesh domains, controlling the maximum scallop height between adjacent spirals, and optimizing the mapping origin to ensure uniform scallop distribution. We address these by integrating surface flattening, binary search-based spacing control, and a perturbation-driven functional energy metric quantifying scallop uniformity. The optimal origin is computed via gradient descent on this energy. Extensive validation on complex models – including low-quality and high-genus meshes – demonstrates the robustness and generality of our approach. Experimental results demonstrate that the proposed method improves the overall balance among machining efficiency, motion stability, and surface quality. Compared with the decomposition-based PDE method, it reduces machining time by 14.23%, suppresses actuator impact, and improves surface-quality consistency. Compared with the iso-planar method, it avoids tool lifting and re-entry, reducing machining time by 6.51%, maximum acceleration by 14.96%, and the most severe negative runout by 16.61%.