Catmull–Rom spline-based C3 continuous tool path smoothing method for five-axis CNC milling

Abstract

Tool-path smoothing is essential for ensuring continuous motion at transition corners between linear segments, since kinematic discontinuities degrade both machining efficiency and surface quality. Most existing spline-based methods achieve only G² or C² continuity and therefore produce discontinuous jerk, which can excite high-order structural resonances. Achieving true C³ continuity remains challenging, particularly because synchronizing tool-tip position and orientation is complicated by the nonlinear relationship between arc length and spline parameterization. This study presents an analytical Catmull–Rom (CR) spline–based method for C³ continuous tool-path smoothing in five-axis CNC milling. Transition corners are replaced by adjustable Catmull–Rom (ACR) splines, whose control points and tuning parameters are designed or optimized to constrain the deviation from the original path. The remaining linear segments are also substituted with ACR splines to enforce position–orientation synchronization, with control points that can be chosen analytically to guarantee zero synchronization error. The proposed method is fully analytical and non-iterative. Numerical simulations demonstrate that the generated tool paths satisfy prescribed geometric tolerances, produce smooth and continuous jerk profiles, and achieve exact synchronization between tool-tip position and orientation.