# Research on Multiple-Axis Contour Error Suppression Method Based on Composite Layered Control
## 1 Introduction
To eliminate the influence of external disturbance and ensure the stability and robustness of the system, this sliding-mode control paper must set the switch gain to be relatively large, and the increase in the switch gain will lead to severe flutter, so the design of the switch gain must change with the change in the external disturbance to obtain a minimum gain that can suppress disturbance. However, in practical applications, it is challenging to determine the upper and lower bounds of external uncertainty disturbances, and even if the type of disturbance can be fully understood, the switch gain cannot be adjusted in real time. Therefore, while improving the sliding-mode approach law, a disturbance observer is introduced to estimate and compensate for uncertain external disturbances, thereby improving the accuracy of single-axis positioning while suppressing flutter phenomena. Based on enhancing the sliding-mode disturbance observation controller, the position error information of each axis is introduced into the cross-coupling controller, and mutual correction is carried out through contour error compensation to reduce the contour error caused by the independent information of each axis. In the above literature, although the single-axis positioning accuracy has been improved, the inter-axis collaborative motion error has yet to be considered. [PERSON] [13] first proposed the CCC control method in a paper in 1980, and its effectiveness was experimentally demonstrated in a dual-axis control system. The essence of CCC comprises two parts: contour error estimation and contour error suppression. Therefore, it has received extensive research from scholars in contour error suppression algorithms. For example, [PERSON] et al. [27] proposed a PID cross-coupling control method based on the position domain, which utilizes the advantage that the reference axis in position domain control does not affect the overall contour error of the system, not only improving the system contour accuracy but also strengthening the robustness of the controller. [PERSON] [28] et al. proposed a control strategy that combines feedforward, feedback, and CCC controllers, and they used a second-order estimation method for contour error estimation. Finally, the ability of this method to reduce contour error was verified on a three-axis experimental platform, which was superior to traditional CCC. [PERSON] [29] et al. proposed a type of iterative learning CCC algorithm for biaxial systems and selected a contour error model based on an algebraic equation. The experimental results show that the more the algorithm cycles, the better the contour error suppression effect. In recent years, there have been several outstanding papers on contour error suppression. A study by [PERSON] [30] et al. delves into the precise motion of feed drive axes, employing an in-depth analysis of helical contour movements. They propose an iterative-learning-based contour error suppression method based on helical cycles, significantly enhancing the accuracy of thread manufacturing. However, a notable drawback lies in the increased computational load due to the inclusion of iterative learning methods. On the other hand, [PERSON] [31] et al. present a method combining two-degree-of-freedom PID and compensators, minimizing tracking errors for each axis and simultaneously reducing dual-axis contour errors. Nevertheless, this paper overlooks the complexity of control design issues and fails to address contour error problems within a specific single plane in spatial contours. In today's research on contour error control, most papers focus on controller designs aimed at enhancing overall contour precision. There is a common assumption that individual axes in machine tool processing are of equal significance. However, in actual machining processes, the coupling relationships between axes vary, and the contour precision differs across different planes when facing diverse accuracy requirements.