## 2 Spatial Similarity Analysis and Sampling Method of Robot Errors in Configuration Space
### Experimental Analysis of Robot Error Spatial Similarity in Configuration Space  
A measurement system is set up to verify the spatial similarity distribution characteristics of robot errors, as shown in Figure 3, and a series of comparative experiments is conducted. A KUKA KR300 R2500-ultra robot (KUKA AG, Augsburg, Germany) equipped with a hole-making end-effector with a weight of 120 kg is used and an FARO Vantage-E6 laser tracker (FARO Technologies Inc., Orlando, FL, USA) is employed to measure the real TCP positions of the end-effector. During the experiments, the spatial similarity distribution of robot errors is verified under two conditions: when the robot is unloaded or loaded and when the end-effector orientation is fixed or changed. A 1.5 inch Spherical Mounted Retroreflector (SMR) (FARO Technologies Inc., Orlando, FL, USA) is mounted on the end of the robot, with its center defined as the TCP. Before the experiment, the base coordinate system {B} and tool coordinate system {T} of the robot are measured and calculated using the laser tracker. In the measurement of {B}, the A1 and A2 axes are rotated sequentially, and the trajectories of the TCP are measured using the laser tracker. Then, the Z-axis of the {B} system is defined, aligning with the A1 axis, and the X-axis is defined perpendicular to both the A1 and A2 axes, based on the coordinate system definition shown in Figure 1. In the measurement of {T}, the A5 and A6 axes are rotated sequentially, and the trajectories of the TCP are measured and the rotation axes are calculated. The X-axis of the {T} system is defined, aligning with the A6 axis, the Z-axis is defined perpendicular to both the A5 and A6 axes, and the SMR center is defined as the TCP. A rectangular region of 1 m \(\times\) 1 m \(\times\) 0.5 m is selected within the robot task workspace for experimental measurement of the robot error distribution in Cartesian space, and 50 points are randomly chosen within this region, with robot end-effector orientations fixed. A range of joint angles is also selected for experimental measurement of the robot error distribution in configuration space with the following ranges: A1 Range = [\(-5^{\circ}\), \(5^{\circ}\)], A2 Range = [\(-85^{\circ}\), \(-95^{\circ}\)], A3 Range = [\(85^{\circ}\), \(95^{\circ}\)], A4 Range = [\(-5^{\circ}\), \(5^{\circ}\)], A5 Range = [\(85^{\circ}\), \(95^{\circ}\)], A6 Range = [\(-5^{\circ}\), \(5^{\circ}\)]. Fifty joint angles are randomly selected as the measurement set within this sampling space, ensuring that the robot end-effector orientations are varied. The load status of the robot is considered in two scenarios: unloaded and under 120 kg load. During the experiment, depending on whether the robot pose is fixed or varied, a sample point list of the fixed or varying pose is selected, and the robot is controlled to move to each position sequentially for the real TCP position measurement. After the measurements, for each sampling point, the theoretical TCP position is calculated based on the joint angles using Equation (3), and the robot error is then determined as the difference between the theoretical TCP position and the real TCP position. The spatial similarity of robot errors is analyzed based on the measurement data in both configuration space and Cartesian space. In the scenario where the robot is unloaded and the pose is fixed, the spatial correlation of robot errors in configuration space can be represented by the distribution of _De_ on _Dq_, as shown in Equation (5), and the scatter plot of _De_ on _Dq_ is illustrated in Figure 4a. When _Dq_ is small, the value of _De_ is also small, with a tightly clustered distribution, indicating that, when the joint angles between two points are close in configuration space, the robot errors  
Figure 3: Experiments for the spatial similarity characteristics of robot errors where (**a**) the robot is loaded with a 120 kg end-effector and (**b**) the robot is in an unloaded state. are also similar.