High-Speed Precision Machining and Surface Roughness Determination of Freeform Curves Using Galerkin-NURBS Interpolation and Jerk-Limited Trajectory Planning

High-speed machining of complex freeform geometries faces fundamental challenges in balancing computational efficiency, kinematic constraints, and precision, particularly in high-curvature regions where traditional interpolation methods suffer from geometric errors and jerk-induced vibrations. This study presents a Galerkin-NURBS interpolation framework that integrates Galerkin projection to optimize NURBS parameterization, minimizing geometric approximation error, and couples it with a jerk-limited S-curve trajectory planning algorithm that enforces C3 continuity while respecting feedrate, acceleration, and jerk constraints. Numerical simulations and machining experiments were conducted on butterfly-shaped and horse-shaped curves using a five-axis CNC machine equipped with rotary/linear encoders and validated via profilometer-based surface roughness measurements. The proposed method achieved a 32.9% reduction in processing time (3.091 s) compared to the CQSF method (4.61 s) and a 35.1% reduction in interpolation steps relative to FSRC. Surface roughness (Ra) values ranged from 0.1271 μm to 0.2009 μm, with most measurements compliant with ISO 21920-1:2021; the maximum value (0.2009 μm) represents the upper bound of the standard’s high-precision threshold for aluminum alloy 6061. These findings demonstrate that the proposed framework significantly improves machining efficiency and surface quality while maintaining geometric fidelity, making it suitable for precision manufacturing applications where sensor-guided process optimization is critical.