# A Self-Calibration Method for Robot End-Effector Using Spherical Constraints
## 1 Introduction  
Due to factors such as the robot's inherent precision limitations and end-effector deformation, there is always a deviation between the theoretical and actual poses of the robot's end-effector coordinate system. In many industrial applications involving robots, high positioning accuracy of the end-effector is essential. Therefore, it is crucial to develop a fast and efficient method for autonomous calibration of the robot end-effector [1, 2]. Calibrating the robot end-effector coordinate system using physical constraints offers advantages in terms of simplicity and speed. The core concept of this method is to constrain the robot's end-effector in various poses at specific points or planes and establish constraint equations based on known physical parameters. By comparing the robot's theoretical pose with its actual pose, errors in different configurations can be identified. Currently, most calibration methods that utilize physical constraints are primarily classified into plane constraints, point constraints and distance constraints. Based on plane constraints, [PERSON] et al. [3] developed a calibration model using a contact measurement probe to measure three planes of the calibration block, thereby completing the identification of the robot's kinematic parameters. [PERSON] et al. [4] extended the manipulator arm with a trigger probe to record the joint sensor values when the manipulator endpoint touched the constraint plane, enabling kinematic error identification. [PERSON] [5] developed a point constraint method by mounting a laser on the robot's end-effector and ensuring that the laser's projection remained fixed on the observation plane, constructing a virtual kinematic chain. This method offers a simple experimental setup and high accuracy in parameter identification. [PERSON] et al. [6],focusing on large industrial robots in the aerospace sector, used a laser tracker to measure the actual position of a target ball fixed to the robot's end-effector, compared it with the theoretical position and calculated error parameters. [PERSON] et al. [7] used a 3D displacement measurement device to constrain the center point of the robot's end-effector to a fixed point. By altering the robot's poses and recording the corresponding joint angles, they achieved an average absolute positioning error of 0.339 mm after robot calibration. Based on distance constraints, [PERSON] et al. [8] achieved TCP position calibration using distance constraints between the center of an optical target ball and the origin of the end-effector flange. [PERSON] et al. [9] designed a self-calibrating movable ball rod comprising two G3 steel balls with known radius and center-to-center distance. By measuring the ball center positions with the robot's end-effector measuring device and comparing these measurements to the theoretical distances, they established distance constraints to perform calibration. End-effector positioning errors in robots can generally be categorized into kinematic errors and non-kinematic errors. Kinematic errors are parameter inaccuracies that occur during the kinematic modeling of the robot. These include errors resulting from joint wear and link deformation due to prolonged use, as well as deformation of robot components caused by environmental factors such as temperature and humidity. In contrast, non-kinematic errors arise from factors external to the robot itself. This category encompasses errors due to flexible deformations of robot links caused by payload effects and deformation errors of the end-effector tool resulting from insufficient stiffness. [PERSON] et al. [10], addressing the effects of the robot's own weight and external loads, developed a flexible error model for the primary stressed joints to enhance the robot's positioning accuracy. [PERSON] [11], considering both joint and link deformation as well as the impact of the robot's weight, established an end-effector deformation model.