for windmills [7], industrial robots are more efficient than machine tools. Machining is an important process from raw materials to finished products [10] that is mainly realized using machine tools with high precision and stability. Although industrial robots have advantages, their low stiffness characteristics can cause unexpected deformations and vibration/chatter during the machining process [11; 12]. Two types of industrial robots, parallel robots and serial robots, are being investigated for machining. A parallel robot is a type of closed-loop structure (Figure 1a), which is composed of a mobile platform connected to a fixed base by a set of identical or different parallel kinematic chains (legs) [13]. A serial robot is a type of open-loop structure, as shown in Figure 1b. It is constituted by a mechanical system consisting of robust links (arms) coupled with either rotational or translating joints. Different tools can be fixed on the flange for machining tasks. Due to the closed-loop structure, the parallel robot has the characteristics of high stiffness and position accuracy compared to the serial robot. However, this structure limits the robot's operational workspace and flexibility. As a result, more research has been conducted to improve the stiffness of the serial robot. In the machining process, a cutting force of 500 N introduces a position error of 1 mm to a serial robot, while the error introduced to a CNC machine tool is less than 0.01 mm [4; 14; 15]. The experimental results indicate that the static stiffness of industrial serial robots is in the range of 10\({}^{5}\)-10\({}^{6}\) N/m [16; 17; 18]. In addition, the stiffness changes as the robot's posture configuration changes, resulting in a nonlinear distribution throughout the workspace, which leads to inaccurate machining [19]. Therefore, most industrial robots operate in fields with low precision requirements and no external contact forces. The number of industrial robots in high-value-added, high-precision industries is far lower than the number of CNC machine tools. The stiffness of industrial robots depends on a variety of factors. The primary sources are elastic deformation caused by the geometric and material properties of various parts, including the joints, connecting links, bases, actuators, and other transmission components of robots. Furthermore, the active stiffness provided by the position control system of  
Figure 1: Two types of industrial robots for the machining process: (**a**) a hexapod parallel robot and (**b**) a KUKA six-axis serial robot. robots requires consideration [20; 21; 22; 23]. Joint deformation results in most errors. Under the weight or wrench of a robotic arm, an additional angle increment is generated by the joint deformation. Then, the angle increment \(\Delta\theta\) multiplied by the long-length mechanical arm would result in serious errors in the end effector (EE) and reduce the precision and machining quality. Therefore, traditional research usually ignores other deformations and focuses solely on joint deformation. Nevertheless, for many engineering applications, other deformation factors cannot be ignored, such as heavy load conditions, hard material removal, and high-temperature operations [24; 25; 26; 27; 28]. * To improve the stiffness of a robot, manufacturers typically increase the cross-sectional area of the arm, which increases the weight of the robot. In addition, large-part production requires the use of large robots. Due to the gravity of the connecting link and the high load on the EE, each joint of the robot has a large deflection, and the running trajectory of the EE has a large deviation [27; 29; 30]. * Significant variation in robot stiffness affects the stability of the machining process. In this case, machining chattering problems are particularly prominent [6; 31; 32; 33]. * High temperatures usually cause thermal deformation and expansion in robot parts. Model parameter errors can occur during the stiffness identification process [28].