###### Abstract
However, it requires iterative computation of new control points, which is inefficient and alters the original curve shape. [PERSON] et al. [9] proposed a B-spline fitting scheme taking advantage of the B-spline's strong convex hull property. In this process, the knot vectors and control points of curve segments, exceeding the prescribed tolerance for chord errors, are refined analytically. The optimization technique [10] fully utilized the tolerance band to minimize the maximum curvature of a B-spline curve with seven control points by adjusting the positions of two control points. However, this approach was specifically employed for local transitions. [PERSON] et al. [11] proposed a novel approach for adaptively discretizing lengthy linear tool-paths to confine nonlinear interpolation errors. Nevertheless, the existing error models are not applicable to industrial robots. Recently, [PERSON] et al. [12] constrained the nonlinear errors in the path in joint space, achieving motion smoothness. However, the drawback is that the errors in this approach cannot be analytically controlled. By using curvature scanning to obtain critical points on the curve and dividing the curve into multiple subcurves [13], it cannot be guaranteed that the chord error in the segmented curves satisfies the given constraints. Secondly, there is a need for real-time trajectory generation technology that calculates robot trajectories satisfying kinematic constraints for each axis. Known constraint parameters are required for feed rate scheduling. Ref. [14] proposes a relatively comprehensive S-shaped feed rate scheduling algorithm (CSFA) with restrictions on jerk, acceleration, and feed rate. It uses a dynamic forward-backward scanning strategy with restrictions on chord error, normal/tangential acceleration, jerk, and feed rate. However, in the error compensation part, a segmented rounding method is used. Although it eliminates the error in segmented arc length interpolation, it introduces feed rate fluctuations at adjacent spline connection points and does not consider robot joint constraint issues. Since the bidirectional scanning cannot guarantee the success of dynamic forward scheduling, ref. [15] introduces the feed rate bisection method to search for feasible boundary velocities. Although it can solve the problem of failed dynamic forward scheduling, its execution efficiency is severely reduced. To compensate for the limitations of unidirectional S-curves in NURBS curve feed rate scheduling applications, [PERSON] et al. [16] proposed a bidirectional scanning algorithm that obtains feed rate parameters for key points and considers the rounding error and its compensation strategy to improve interpolation accuracy. However, this method does not consider the constraint of a contour error in the middle part of the curve segment, resulting in the speed exceeding limits in the planned curve. To consider the continuous existence of constraints, it is essential to use smooth velocities for trajectory planning. Although the trapezoidal feed rate scheduling method is simple to implement [17], the discontinuity in acceleration can cause vibrations during frequent acceleration and deceleration, leading to a decrease in the lifespan of the motion equipment. The G1 and G2 discontinuities occurring at the junctions of two linear segments inevitably result in frequent feed rate fluctuations and acceleration variations [18]. Therefore, toolpath smoothing operations are crucial in numerical control systems to improve path continuity [19]. To address this problem, many feed rate scheduling methods that restrict acceleration jumps have been proposed, such as the time-optimal method [20; 21], the sine-shaped method [22], and the S-shaped method [23; 24]. [PERSON] et al. [25] designed a systematic approach to design time-optimal S-curve motion profiles. [PERSON] et al.