## 1 Introduction  
A thread milling operation is a machining process in which the feed drive axes of a machine tool simultaneously move a thread milling tool along a helical path to perform a cutting operation. Therefore, the helical contour motion accuracy of feed drive axes while performing thread milling operations becomes crucial. In recent decades, owing to the importance of the contour motion accuracy of computer numerical control (CNC) machine tools, many contour control designs have been proposed to improve the contour motions of feed drive axes. [PERSON] et al. [1] proposed a velocity-based contour control method that used a B-spline-based wavelet neural network to approximate unknown dynamics and used neural network identification to design a robust control to improve the contour motion of dual-linear-motor-driven gantry stages. [PERSON] et al. [2] transformed the desired contour path and tool orientation in the workpiece coordinate frame to a five-dimensional hypercube in a machine coordinate frame and then adopted the equivalent error method to design a contouring controller to reduce the errors of the contour path and tool orientation. [PERSON] et al. [3] compensated weighted contour error components to velocity commands in the robotic task space of a six-degrees-of-freedom robot arm. They proposed a component-based contour control mechanism that can improve the contour motion accuracy of the robot arm performing machining tasks. [PERSON] et al. [4] integrated generalized predictive control and feedback correction algorithms to develop a real-time contour error control design for five-axis machine tools. [PERSON] et al. [5] improved the conventional cross-coupled control design to increase the contour accuracy of biaxial motion control systems. [PERSON] et al. [6] adopted a neural memory network to develop a contour error estimation and compensation method for CNC systems. [PERSON] et al. [7] designed a five-axis cross-coupling control method in joint space to simultaneously reduce task-space and joint-space contour errors. [PERSON] et al. [8] referred to the calculations of distance and contour errors and adopted the trajectory compensation method with gain adjustment to modify the reference trajectory for improving the contour control performance of motion control systems. [PERSON] et al. [9] considered the input and output data of a two-dimensional linear motor system while performing reciprocating motions and then proposed a model-free adaptive control algorithm that can improve the contour motion accuracy of the linear motor system. [PERSON] et al. [10] proposed a discrete-time fractional-order sliding-mode contour error control design based on fractional-order sliding surface stability analysis and linear matrix inequality. [PERSON] et al. [11] proposed an equivalent-plane cross-coupling control design to apply the developed biaxial cross-coupling control to three-axis contour error control. [PERSON] and [PERSON] [12] transformed a three-dimensional contour coordinate frame into a two-dimensional contour coordinate frame represented by radial and angular axes and then adopted the computed-torque control and proportional-derivative control to design a motion controller. [PERSON] et al. [13] proposed a contour error control design based on the generalized extended state observer and model predictive control to reduce the contour errors of networked multi-axis motion systems caused by network-induced delays. Many contour control designs that can effectively improve the contour motion accuracy of the feed drive axis have been proposed; however, most of them focus on improving contour motion properties, but the adverse influence of external disturbances on the feed drive axes has yet to be carefully discussed. Particularly, the cutting force in the thread milling operations affects the contour motion accuracy of feed drive axes while performing a helical path.