# Research on Multiple-Axis Contour Error Suppression Method Based on Composite Layered Control
## 1 Introduction  
When it comes to contour error suppression in multi-axis high-precision servo control systems, our considerations extend beyond merely optimizing the system's performance. We are contemplating its broader role in industrial automation, advanced manufacturing, and digital production. In the industrial landscape, the development and application of equipment such as CNC machining tools, multi-DOF robotic arms, and precision optical instruments [1; 2; 3; 4; 5] have consistently driven technological progress in manufacturing. The efficiency, precision, and reliability of these devices are pivotal for the industry's advancement, and the effective suppression of contour errors directly impacts product quality, production efficiency, and the overall manufacturing process optimization. By effectively reducing contour errors, we can enhance product accuracy and quality and decrease waste, thereby saving costs and enhancing competitiveness. This holds significant implications across various industries, including aerospace, automotive manufacturing, and medical devices, among others. Moreover, with the evolution of Industry 4.0 and smart manufacturing, multi-axis high-precision servo control systems play an increasingly crucial role in factoryautomation and digitalized production. Optimizing contour error suppression algorithms is not just about enhancing product quality; it is a pivotal driver of smart manufacturing. In multi-axis servo motion systems, the accuracy of single-axis positioning and the synergy between axes are closely related to the overall contour accuracy. Before suppressing the general spatial contour error, it is necessary to ensure the positioning accuracy of each axis and then consider the dynamic characteristics and parameter matching issues between axes [6; 7]. In recent years, scholars have conducted extensive research on the accuracy of single-axis positioning, such as feedforward control [8; 9], active disturbance rejection control [10], adaptive control [11], etc. In the literature, although the positioning accuracy of each axis has been improved by designing a single-axis controller, we know that the contour error accuracy in multi-axis servo control systems is not only affected by the single-axis control accuracy but also by the dynamic matching characteristics between each axis, which affect each other and thus affect the overall contour accuracy [12]. More than simply improving the accuracy of single-axis positioning is required to enhance the contour accuracy further. Therefore, [PERSON] proposed the cross-coupled control (CCC) method in 1991 [13]. In CCC, contour error estimation is based on the spatial geometric relationship between single-axis tracking error and contour, and then allocation compensation is carried out through the cross-coupled controller. The CCC method is considered the most effective control method to reduce contour error and is often used in combination with other control methods, such as adaptive control [14], sliding-mode control [15], predictive control [16], iterative learning control [17], repetitive learning control [18], etc. Among them, sliding-mode control [19; 20] has become a research hotspot amongst the current control methods because of its insensitivity to system model parameter perturbation and its strong robustness, and it is suitable for use in complex and strongly nonlinear multi-axis control systems. Examples of sliding-mode control applications in motors can be found in references [21; 22; 23] and others. However, sliding-mode control is not flawless. Due to the presence of the sign switch function, traditional sliding-mode control generates high-frequency chatter when the controlled state approaches the stable point of the system in practical use. Therefore, scholars of sliding-mode control have proposed many improved methods to suppress chatter, such as the reaching law [24], the high-order sliding-mode method [25], and the non-singular terminal sliding mode [26].