SO(3)-based orientation smoothing for improved rotary-axis motion in five-axis machining

In five-axis CNC machining, rotary axes may exhibit abrupt and unstable motions near kinematic singularities, leading to degraded machining accuracy and potential machine damage. This issue occurs because smooth variations in tool orientation do not necessarily result in smooth A-C rotary-axis motions. To address this challenge, we propose an approach where tool orientations are treated as elements of the Special Orthogonal Group SO(3), with smoothing performed directly within this group under machining tolerance constraints. The approach is guided by a pullback-metric interpretation, which clarifies why direction-based representations may lose continuity near singular postures. The smoothed orientations are subsequently converted to A–C commands through inverse kinematics and refined locally to reduce rapid variations in the rotary axes. Numerical studies on benchmark trajectories, alongside comparisons with several representative singularity-avoidance strategies, demonstrate that the proposed formulation effectively mitigates sharp variations typically observed near singular postures. The resulting rotary-axis motions exhibit significantly reduced fluctuations in velocity, acceleration, and jerk. Experimental validation, conducted on a standard S-shaped surface, confirms the effectiveness of the approach in practical machining scenarios.