# Modeling and Compensation of Motion Errors for 6-DOF Robotic Manipulators
## 1 Introduction  
Since the advent of robots in the 1940s, robot technology has constantly improved. After nearly a century of development, this technology has become an indispensable part of industries and production, especially in the field of precision machining and testing [1,2]. However, due to their series structure configuration, robotic manipulators have some problems, such as jitter and a relatively large end error caused by insufficient stiffness. To improve the stiffness and working performance of robots, many researchers have carried out relevant research during the past few decades. [PERSON] [3] obtained an estimate of each joint stiffness of an industrial robot by analyzing the joint rigidity of the mechanical transmission system structure and parameters of conversion. Based on the analysis of joint stiffness and arm stiffness, [PERSON] [4] employed the error model to establish the stiffness model of industrial robots. In 2011, [PERSON] et al. [5] from France established the static stiffness model of industrial robots and proposed the relevant joint stiffness identification method with the condition that the complementary stiffness matrix term and infinite stiffness of the robot connecting rod were disregarded. In 2013, [PERSON] et al. [6] from Russia used the data measured by sensors inside the robot to identify the joint stiffness of the robot. [PERSON] et al. [7] pointed out that the stiffness at the end of a robot would change with its posture. In 2015, [PERSON] et al. [8] in France converted the flexibility of the arm to the flexibility of joints, established the robot stiffness model, and then identified the stiffness parameters by considering the flexibility of the manipulator and joints. In 2018, [PERSON] et al. [9] proposed the concept of an offset stiffness matrix, explored the impact of the offset stiffness matrix on milling, and verified its correctness. [PERSON] et al. [10] proposed a vibration control method using a tunable mass damper with variable stiffness, which could avoid the maladjustment effect and suppress the vibration. For the space stiffness of a tandem manipulator composed of traditional joints, much research has been conducted. [PERSON] et al. [11] utilized the bionic method of the human spine to calculate the stiffness matrix of a 6-DOF manipulator to improve its accuracy. [PERSON] et al. [12] proposed a continuous manipulator driven by a variable stiffness wire that could be passively adapted to collision and ensure operation accuracy. [PERSON] et al. [13] proposed a region-based tool path generation method to improve the stiffness in the robot milling of freeform surfaces. Based on the matrix structure analysis method, [PERSON] et al. [14] expounded the calculation method of the Cartesian stiffness matrix of the manipulator and introduced a new performance index of the manipulator. [PERSON] et al. [15] proposed an adaptive impedance force control method for manipulators based on a fuzzy neural network and estimated the stiffness coefficient of the contact environment. [PERSON] et al. [16] combined the Lagrange method with the dynamic stiffness matrix method to build a CDPR vibration model that could capture the dynamic characteristics of cables. In 2020, researchers from Tsinghua University used the deformation coupling method to describe the nonlinear deformation of the manipulator and then established a dynamic model of the arm variable stiffness by combining the assumed modal method and Lagrange equation [17]. [PERSON] et al. [18] proposed a calibration method of robot geometric parameters based on laser tracker measurement, which improved the absolute positioning accuracy of cooperative robots, the D-H parameter error of which is identified and compensated for. [PERSON] et al. [19] corrected the absolute positioning accuracy error caused by the internal mechanism deviation of the robot by modifying the parameters in the underlying controller, so as to improve the operating position accuracy of the robot.