## 4 Bidirectional Feed Rate Scanning of S-Shaped Profile
\[B_{i}(u)=B_{i}(f_{i}(\mu))=\{x_{i}(f_{i}(\mu)),y_{i}(f_{i}(\mu)),z_{i}(f_{i}( \mu))\} \tag{25}\]

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]0

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]1

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]2

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]3

Therefore, the relationship between the derivatives of NURBS curves and the derivatives of Bezier curves is:

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]4

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]5

Utilizing the composite Simpson's rule [34] to calculate the arc length can be perceived as a continuous iterative process for determining the value of the parameter \(u\). This is achieved by computing the parameter \(u_{i+1}\), which corresponds to the next position coordinate of the interpolation cycle, based on the current position's corresponding parameter \(u_{i}\). Derivatives of the parameter \(u\) with respect to time t can be derived, and by expanding

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]6 Bidirectional scanning case classification. these first-order derivatives into a second-order Taylor series, a second-order [PERSON] direct interpolation formula [35] based on feed rate can be obtained:

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]7

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]8

\[A=\left[a\tau+\frac{v^{2}}{\rho}\mathbf{n},0\right] \tag{20}\]9

Figure 6:0

Figure 6:1

Figure 6:2

where \(u_{k}\) is the curve parameter obtained from the previous interpolation step, and \(T\) represents the interpolation period. \(\Delta s_{i}\) is the arc length increment in the current interval. The interpolation is directly computed using the anticipated increment \(u\) without any compensatory measures, categorizing it as an open-loop Taylor direct interpolation. To address the aforementioned issues, initially, the second-order Taylor expansion is employed to calculate the initial value of the parameter \(u\). Subsequently, an optimization model, considering both the arc length and parameters, is established:

Figure 6:3

Figure 6:4

The entire system flow is illustrated in Figure 6:5

Figure 6:6 Interpolation process of robot NURBS curve motion. ## 6 Experiment and Results

In order to verify the algorithm's robustness, two verification experiments are carried out in this section. The first one is used to verify the effectiveness of the high-precision NURBS interpolator proposed in this paper, and the second one is used to verify the practicality of the proposed robot tool tip and attitude synchronous interpolation. ### NURBS Interpolator for Butterfly Curve

In this section, simulations are conducted to show the effectiveness of the proposed method. The computer utilized has a CPU: I7-11800H-2.30 GHz (Intel Corporation, Santa Clara, CA, USA), 64 G-3200 MHZ memory, and the MATLAB (The MathWorks, Natick, MA, USA) version is 2022b. The plane butterfly NURBS curve (see Figure 9(a)) is employed as the test subject for algorithm evaluation. The chord error is set as \(\delta=0.001\) mm. Figure 9(b) shows the butterfly curve segmented into a type-C curve by the zero curvature point, and Figure 9(c) shows the butterfly curve segmented into 1579 type-C curves that satisfy \(\delta=0.001\) mm. Each Bezier curve is distinguished by color. In Figure 9(d), with the chord error, the chord value is less than 0.001 mm, while the chord is a hundred times higher when without chord error control. In Figure 9(e),f, respectively, the displacement and feed rate of the bidirectional feed rate scanning of the S-shaped profile are shown. Based on these figures, the plant area always has a longer Bezier curve, while the peak part has a shorter curve, which makes sense. Figure 6:7 Cont. Systematic comparisons are implemented between the proposed method and the other methods (L.H. [8], X.D. [14], L.F. [32]) mentioned in this paper.