###### Abstract
[26] studied the asymmetric motion properties of feed rate scheduling, as traditional or most feed rate planning methods rely on symmetric formulas for the start and stop segments, but the time is not optimized in those cases. Thirdly, for the parametric solution of NURBS curve interpolation, [PERSON] et al. [27] performed a first-order Taylor expansion treating the interpolation parameters as a function of time, but the calculation precision is not high. To improve the calculation precision, [PERSON] et al. [28] proposed an interpolation point parameter solution algorithm based on second-order Taylor expansion, but it requires a large amount of computation, which impacts the real-time capability of the algorithm. [PERSON] [29] proposed that within each interpolation period, the maximum feed rate is determined by the specified chord error and constant material removal rate. In existing research [30; 31], it has been demonstrated that the Taylor expansion method can generate trajectories identical to the reference method, but feed rate fluctuations are inevitable in regions with high curvature. An optimization algorithm [32] is proposed to improve the precision of NURBS curves. This algorithm employs an iterative process to calculate block displacement, with the NURBS parameter u as the optimization target. However, the requirement to repeatedly compute the accurate value of u results in the inability to perform online interpolation. Most interpolation methods use Taylor expansion to approximate the next interpolation parameter. In response to the above issues, this article proposes a tool speed scheduling and high-precision interpolation system for industrial robots for NURBS curves. Firstly, a fast and accurate calculation method for tool path curvature is proposed. The knot insertion technique is introduced to segment the original tool path into segments based on curvature, which creates a set of rational Bezier curves. The entire processing curve is ensured to meet the requirements of the chord error. Secondly, considering the impact of the nonlinear kinematics of robot joints, a dynamic optimization of the maximum feed rate and maximum acceleration is conducted during the bidirectional scanning process of the traditional feed rate extrema curve. This optimization achieves dynamic speed regulation and braking functions within an adaptive S-shaped feed rate profile. The preset values for the maximum feed rate and maximum acceleration constraints in bidirectional feed rate scanning are adjusted, ensuring smooth robot interpolation while adhering to joint kinematic constraint. Finally, during the curve interpolation stage, a compensation interpolation strategy using Taylor expansion and arc length parameterization is applied to the segmented NURBS curve set. The parameter remapping method is used to ensure parameter continuity at the junction points of adjacent curve segments, without transferring all the original NURBS control points to the underlying structure. This reduces storage capacity and eliminates nonlinear errors in interpolation. The remaining sections of the paper are organized as follows. Section 2 proposes a method for segmenting NURBS curves into a set of rational Bezier convex curves based on a closed-loop chord error constraint. Section 3 introduces an improved adaptive S-shaped feed rate scheduling algorithm, which achieves an adaptive planning mode under a given displacement. It also presents a method for robot speed control and braking planning under arbitrary states, enabling smooth speed regulation and braking based on S-shaped feed rate scheduling. Section 4 proposes a bidirectional feed rate profile scanning algorithm that effectively combines robot joint constraints to meet maximum performance requirements. Section 5 addresses the problem of the continuous interpolation of convex curve sets and presents a parameter remapping method based on Taylor expansion to ensure parameter continuity at the junction points of adjacent curves.