## 1 Introduction  
Recently, robots have been increasingly used in the field of mechanical machining [1, 2], especially in scenarios that require high precision, high efficiency, and flexibility, such as for aerospace assembly [3, 4] and metal and composite material milling [5, 6, 7]. However, the typical positioning accuracy of the robot is merely \(\pm 1\) mm [8] with the control method in the original controller provided by the robot company, which is difficult to satisfy the needs of high-precision machining tasks. Robot error compensation technology is an effective measure to cope with this problem. Error compensation refers to the technology using measuring equipment combined with certain numerical algorithms to reduce or eliminate the positioning error of an industrial robot. Robot error compensation can be categorized into two categories: online and offline compensation. The former needs to rely on a sensor such as a laser tracker to measure the actual positions of the robot's TCP online and then to compensate for the TCP error in real-time. Electroimpact Inc. [3, 9] developed an accurate industrial robot by attaching the optical encoder to each joint of the robot and customizing the Siemens numerical control system with real-time compensation algorithms. Modification of the robot joints and the control system resulted in an enhancement of the positioning accuracy from \(\pm 2\)-\(4\) mm to \(\pm 0.18\) mm with no restriction on tool orientation. [PERSON] [10] proposed methods to correct the path error of a robotic machining system by using a laser tracker of three degrees of freedom and reduced the hole position errors to 0.17 mm in a robotic drilling test. Combining a six-degrees-of-freedom laser tracker and a customized industrial robot with an optical encoder and numerical control system, [PERSON] [11] further improved the static absolute accuracy of the robot to 0.07 mm. Instead of the laser tracker, [PERSON] [12] used an optical coordinate measure machine to correct the error of the robot and reached an accuracy of \(\pm\)0.05 mm for position and \(\pm\)0.05\({}^{\circ}\) for orientation. Although online feedback methods can significantly amend the accuracy of industrial robots, the high dependency on precise measuring instruments during the entire operation time is hardly acceptable in complex industrial sites. The off-line compensation methods are typically divided into kinematic calibration and non-kinematic compensation. Kinematic calibration methods identify robot kinematic parameter errors and the transformation relationship of TCP by measurement results of the Cartesian sampling procedure. [PERSON] [13] mounted three High Modulus Polyester cords on the end of the robot to measure the position of the TCP and calibrate the robot. The result showed an enhancement from an initial average positioning error of 17.88 mm-1.16 mm. [PERSON] [14] proposed a calibration method using three dial gauges and the calibration balls, which reduced the positioning error to within 0.5 mm in the target workspace. [PERSON] [15] compared an in-contact method using a plane constraint and an off-contact method using a 3D motion tracking system, which improved the RMS errors of the robot to 0.2515 mm and 0.156 mm, respectively. Through the laser tracker with higher precision, [PERSON] [16] proposed a calibration method based on the six-degrees-of-freedom pose measuring strategy, which requires the mounting of three reflectors at the end of the robot and reduces the max error of the robot to 0.32 mm. To avoid the transformation error of frames between the measurement device and the robot, [PERSON] [17] calibrated the robot by measuring the distance accuracy and increased the absolute accuracy to 0.27 mm in a single direction. However, the accuracy achieved by kinematic calibration methods still failed to meet the requirements of aerospace manufacturing, which is 0.25 mm for the vast majority of applications[3], because of the neglect of non-kinematic errors.