## 4 Bidirectional Feed Rate Scanning of S-Shaped Profile
The controller's hardware platform is equipped with the SpeedGoat RCP (The SpeedGoat, Bern, Switzerland) real-time simulation platform, featuring a computation cycle of 1 ms. The control system is developed using the MDH model and leverages the real-time functionality of MATLAB Simulink (The MathWorks, Natick, MA, USA). 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 version is 2022b. In Figure 13a, the fan curve segmentation result is presented, employing the chord error constraint method with 1454 segments, and the maximum chord error is 0.001 mm. Figure 13b displays the bidirectional feed rate scanning of the S-shaped profile. The surface of the fan NURBS curve is shown in Figure 13c,d. It can be observed from the figures that the machining surface obtained by the proposed method exhibits no obvious overcutting, indicating a relatively smooth surface quality. The proposed method was validated through a milling experiment using a planar fan NURBS curve path on a robot. The results confirm the correctness and effectiveness of the proposed approach. It is demonstrated that the method can generate smoother machining paths, leading to a smoother tool tip feed rate. Consequently, this significantly enhances both the efficiency and quality of robot machining. ## 7 Conclusions

This paper introduces a high-precision planar NURBS interpolation system for industrial robots. Leveraging knot insertion technology, the global NURBS curve is segmented into multiple rational Bezier subcurves that satisfy the chord error constraint. Subsequently, to address limitations in the current S-shaped profile, a real-time speed adjustment scheduling method is proposed to ensure feed rate adjustments within the lookahead range during

Figure 13: Fan NURBS curve: (**a**) fan curve segmentation result by using chord error segmentation method; (**b**) the feed rate of bidirectional feed rate scanning of S-shaped profile; (**c**) overhead view; (**d**) side view. bidirectional scanning. Additionally, by integrating segmented Bezier parameter remapping theory, the continuity of parameters at the junctions of two segments is preserved. The robustness of the proposed scheme is validated through butterfly NURBS curve simulations, and the effectiveness of the proposed approach is verified by an experiment conducted on an industrial robot using a fan NURBS curve. [PERSON] contributed the central idea and wrote the initial draft of the paper. [PERSON], [PERSON], [PERSON] and [PERSON] contributed to the refining of the ideas and revised this paper. Software and Writing--original draft, [PERSON]; resources and writing--original draft, [PERSON]; writing--review and editing, [PERSON], [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Key Research and Development Program for Robotics Serialized Harmonic Reducer Fatigue Performance Analysis and Prediction and Life Enhancement Technology Research, Grant No. 2017 YFB1300603. All data generated or analyzed during this study are included in this paper or are available from the corresponding authors on reasonable request. The data are not publicly available due to intellectual property protection. The authors declare no conflict of interest. ## References

* (1) [PERSON]; [PERSON]; [PERSON] Optimum kinematic design of a planar cable-driven parallel robot with wrench-closure gait trajectory. _Mech. Mach. Theory_**2016**, 99, 1-18. [CrossRef]
* (2) [PERSON]; [PERSON] Experimental investigation of feed rate limitations on highspeed milling aimed at industrial applications. _Int. J. Adv. Manuf. Technol._**2007**, 32, 1104-1114. [CrossRef]
* (3) [PERSON]; [PERSON] _The NURBS Books_, 2 nd ed. ; Springer: Berlin/Heidelberg, Germany, 1996. * (4) [PERSON]; [PERSON] Tool path generation for freeform surface models. In Proceedings of the Second ACM Symposium on Solid Modeling and Applications, Montreal, QC, Canada, 19-21 May 1993; pp. 419-428.