随着先进制造业的发展，数控平台等数控装备广泛应用于航空航天、汽车等工业制造领域。然而，在加工大曲率、曲率突变等复杂轮廓轨迹任务时，由于伺服系统滞后性和各轴动态特性不匹配等问题，仅通过控制单轴跟踪误差间接提高轮廓加工精度的控制方式，无法满足复杂零件高轮廓加工精度的要求。因此，针对数控平台轨迹跟踪控制方法展开深入研究，对提高零件的轮廓加工精度、进而对提高数控装备竞争力有重要意义。本文以XY数控平台为研究对象，针对XY数控平台在加工大曲率、曲率突变处轨迹的轮廓加工精度差、控制性能不佳的问题，展开减小XY数控平台轮廓误差的研究。具体研究内容如下：

首先对数控平台位置伺服系统的组成架构进行分析，对永磁同步电机（Permanent Magnet Synchronous Motor, PMSM）电压、磁链方程及电机运动方程进行推导，利用矢量脉宽调制技术实现电机转矩最优控制；分析机械传动系统结构，建立了数控平台机械传动系统的数学模型，最终基于Matlab/Simulink完成了数控平台位置伺服系统建模，为后续轨迹跟踪控制方法的设计提供理论基础。

其次分析了影响轮廓加工精度的因素，阐明了跟踪误差和轮廓误差的区别。在现有轮廓误差估计算法的基础上，基于优选点近似圆逼近原理，提出了四点双圆逼近的轮廓误差估计（Contour Error Estimation, CEE）方法，该方法采取距离实际加工位置点最近的四个采样点，根据这四点构建两个圆，通过两圆与实际加工位置点之间的几何关系来估算轮廓误差。在数控平台位置伺服系统仿真模型的基础上进行仿真加工，仿真结果表明：进行圆形轨迹仿真时，四点双圆逼近法的轮廓误差估计精度比密切圆逼近法提高了4.832μm；进行四角星轨迹仿真时，基于四点双圆逼近法的轮廓误差估计精度比密切圆逼近法提高了9.94μm，证明了四点双圆逼近法的估计精度更高。

接下来，本文提出了一种基于“单轴遗传算法优化的变论域模糊PID（GA-FPID）+四点双圆逼近CEE+单神经元PID交叉耦合控制器”的轨迹跟踪控制方法。设计了基于遗传算法优化的变论域模糊PID单轴位置控制器，并基于交叉耦合控制原理，设计单神经元PID交叉耦合控制器对轮廓误差进行控制。进行单轴位置控制算法的仿真验证，结果表明单轴GA-FPID控制算法优于PID控制算法，具有较好的响应速度和跟踪性能。对圆形轨迹和四角星轨迹进行仿真加工，仿真结果表明单神经元PID交叉耦合控制器优于PID交叉耦合控制器，对轮廓误差的控制效果更好。

最后搭建了基于“PC+运动控制卡”的XY数控平台，对本文提出的轨迹跟踪控制方法进行实验验证。采用不同的轨迹跟踪控制方法加工圆形轨迹和四角星轨迹，实验结果表明本文的轨迹跟踪控制方法能够有效地减小大曲率轨迹及曲率突变轨迹加工时产生的轮廓误差，体现了良好的轨迹跟踪控制性能。

With the development of advanced manufacturing, computer numerical control equipment, such as CNC platforms, is widely applied in industrial manufacturing fields like the aerospace and automotive industries. However, when dealing with complex contour tracking tasks involving large curvatures and sudden curvature changes, due to problems such as the lag of the servo system and the mismatch of the each axis dynamic characteristics, the control method of indirectly improving the contour machining accuracy only by controlling the single-axis tracking error cannot meet the requirements of high contour machining accuracy for complex parts. Therefore, conducting in-depth research on trajectory tracking control methods for CNC platforms holds significant importance for improving the contour accuracy of machined parts and, consequently, enhancing the competitiveness of CNC equipment. This paper focuses on the XY CNC platform, aiming to address the problems of poor contour accuracy and unsatisfactory control performance when processing trajectories with large curvatures and sudden changes in curvature. Specifically, the research investigates methods to reduce the contour error of the XY CNC platform. The detailed research contents are as follows:

Firstly, the composition and architecture of the CNC platform's position servo system are analyzed. The voltage, flux linkage, and motor motion equations of the permanent magnet synchronous motor (PMSM) are derived, and space vector pulse width modulation technology is utilized to achieve optimal motor torque control. The mechanical transmission system structure is analyzed, and a mathematical model of the CNC platform's mechanical transmission system is established. Finally, based on Matlab/Simulink, the modeling of the CNC platform's position servo system is completed, providing a theoretical foundation for the subsequent design of trajectory tracking control methods.

Secondly, factors affecting contour accuracy were analyzed, and the differences between tracking error and contour error were clarified. Building on existing contour error estimation algorithms, a four-point dual-circle approximation-based contour error estimation (CEE) method is proposed based on the principle of optimal point approximation using circular arcs. This method selects the four sampling points closest to the actual machining position points, constructs two circles based on these points, and estimates the contour error through the geometric relationship between the two circles and the actual machining position points. Simulation machining was carried out based on the simulation model of the position servo system of the numerical control platform. The simulation results show that when conducting circular trajectory simulations, the contour error estimation accuracy of the four-point double-circle approximation method is 4.832 μm higher than that of the closed circle approximation method. When conducting the trajectory simulation of four-pointed stars, the contour error estimation accuracy based on the four-point double-circle approximation method is 9.94μm higher than that of the close circle approximation method, which proves that the estimation accuracy of the four-point double-circle approximation method is higher.

Next, this paper proposes a trajectory tracking control method based on "Variable domain Fuzzy PID optimized by genetic algorithm (GA-FPID) + Four-point double-circle approximation CEE+ single-neuron PID cross-coupling controller". A variable domain fuzzy PID single-axis position controller optimized by genetic algorithm was designed. Based on the principle of cross-coupling control, a single-neuron PID cross-coupling controller was designed to control the contour error. The simulation verification of the single-axis position control algorithm was carried out. The results show that the single-axis GA-FPID control algorithm is superior to the PID control algorithm and has better response speed and tracking performance. The circular trajectory and the quadrantal star trajectory were simulated and processed. The simulation results show that the single-neuron PID cross-coupling controller is superior to the PID cross-coupling controller and has a better control effect on the contour error.

Finally, an XY numerical control platform based on a "PC+ motion Control card" was built to experimentally verify the trajectory tracking control method proposed in this paper. Different trajectory tracking control methods were adopted to process circular trajectories and quadrangular star trajectories. The experimental results show that the trajectory tracking control method proposed in this paper can effectively reduce the contour errors generated during the processing of large curvature trajectories and trajectories with sudden curvature changes, demonstrating excellent trajectory tracking control performance.