Third-Order Peak Localization of Tool-Axis Orientation Error for NC Densification in A–C Five-Axis Machining

Linear rotary-axis interpolation in five-axis machining can induce nonlinear tool-axis orientation error despite rotation-tool-center-point compensation of the tool-tip position. This study analyzes this effect for well-conditioned blocks of an A–C rotary-table configuration using the signed out-of-plane deviation from the endpoint geodesic plane on the unit sphere S2. A local asymptotic analysis explains the symmetric second-order midpoint approximation and derives the leading third-order shift of the local error peak. A peak-guided uniform (PGU) method for densifying numerical control (NC) blocks is then developed. The third-order model selects a probing location, at which the exact nonlinear error is evaluated; local quadratic scaling then directly estimates the subdivision number. The reference-path-reconstructed sequence is assessed by independent dense sampling. For the largest representative block, the peak-location discrepancy decreases from 0.072 with the midpoint assumption to 0.023 with the third-order predictor. In the coarsest tested blade-path case (r=4), PGU matches the inserted-point count of dense-search uniform subdivision and uses 27.2% fewer inserted points than the midpoint and peak-corrected refinement baselines. Across the tested tolerances, PGU satisfies the same dense-validation criterion as the reference methods with comparable final block counts. At r=4, its median runtime is approximately 94.6% lower than that of dense-search uniform subdivision.