Local transition smoothing method with C3 continuity and error control strategy oriented to five-axis linear tool paths

To address issues such as low machining efficiency, unstable motion, and difficulty in controlling position and orientation errors caused by the discontinuity of tangent vectors at segment junctions in five-axis linear tool paths, a local transition smoothing method with C3 continuity and a corresponding error control strategy are proposed in this article. The proposed smoothing method enhances the continuity of linear tool paths by constructing two specific quintic B-spline curves at segment junctions. One curve is used to locally smooth the position path, while the other performs local smoothing of the orientation path on the unit sphere. To effectively control position and orientation errors, a geometric strategy is introduced. By applying the law of cosines and Vieta’s theorem, an analytical expression for orientation error is derived, enabling accurate prediction and control of the errors. The geometric derivative continuity conditions at the junctions are also established, ensuring that the resulting tool paths satisfy C3 continuity mathematically and maintain consistency in parameterization between the position and orientation trajectories. Simulation and experimental results demonstrate that the proposed method not only ensures superior path smoothness but also achieves robust, analytically controlled position and orientation errors, both strictly within predefined tolerances. Compared to conventional C2 continuity methods, the proposed method increases machining speed by up to 16% and reduces peak acceleration and jerk by ∼20% and 23%, respectively. These improvements lead to smoother tool motion, enhanced machining stability, and improved surface finish quality.