## 3 Semi-Physical Experiment and Analysis
### Results of Semi-Physical Experiments
#### 3.2.2 Fine Registration by J-ICP  
To evaluate the accuracy of point cloud registration and relative pose measurement of the proposed algorithm, the transformation matrix \(T_{pred}\) of the source point cloud \(P\) was computed using the registration algorithm for a given target point cloud \(Q\). The actual transformation matrix between the two point cloud sets is denoted as \(T_{gt}\). The transformation error \(\Delta T\) from \(Q\) to \(P\) is defined as in Equation (10). \[\Delta T=T_{pred}\left(T_{gt}\right)^{-1}= \begin{bmatrix}\Delta R&\Delta t\\ 0&1\end{bmatrix}  \tag{10}\]  
where \(\Delta R\) is the \(3\times 3\) rotation residual matrix, and \(\Delta t\) is the \(3\times 1\) translation residual vector. Ideally, if \(T_{gt}\) is entirely accurate, \(\Delta t\) should be the identity matrix. To represent the rotation error of the algorithm, we converted \(\Delta R\) into Euler angles according to the \(ZY\)X rotation sequence. The three elements of \(\Delta T\) were taken as the translation error. In order to enhance the accuracy of position estimation results, this study utilized the J-ICP algorithm in conjunction with coarse alignment results to achieve precise target positioning. Position estimation of a non-cooperative target was conducted under translational and rotational motions, with motion constraints set at a moving distance of 570 mm and a total change angle of 57\({}^{\circ}\). The experimental results, depicted in Figure 11, illustrate the motion estimation curves and errors for the target with unknown motion. For spatial motion, the average estimation errors in the XYZ-axis direction for translation were 1.1 mm, 1.5 mm, and 0.25 mm, respectively. Additionally, the average errors for Eulerian roll, pitch, and yaw in the XYZ-axis were 0.045\({}^{\circ}\), 0.019\({}^{\circ}\), and 0.071\({}^{\circ}\) respectively. As illustrated in Figure 12, we compared the fine alignment results obtained by applying three fine alignment methods (J-ICP, ICP, GICP) after coarse alignment. After the satellite model underwent translation and rotation by the slide rail and turntable, point cloud data were registered frame by frame. The XYZ axis translation error measured utilizing the GICP algorithm was (1.36, 2.46, 0.35) mm, while the XYZ axis Euler angle error was (0.112, 0.056, 0.059)\({}^{\circ}\). However, by applying the method proposed above, those could be reduced to (1.07, 1.57, 0.25) mm and (0.071, 0.046, 0.019)\({}^{\circ}\) respectively. By implementing a joint registration algorithm, this research successfully eliminated cumulative errors and achieved improved accuracy in pose estimation parameters. By comparing the performance of the proposed J-ICP algorithm with classical algorithms ICP and GICP in terms of mean translation error and mean rotation error, as shown in Table 2, we found that our algorithm significantly outperformed the classical algorithms. Figure 10: Alignment errors of the RANSAC method for different noise criteria. (**a**) Translation error; (**b**) rotation error. Specifically, compared to the classical ICP algorithm, the average translation error was reduced by approximately 48%, 56%, and 54%, while the average rotation error was reduced by approximately 56%, 48%, and 65%, respectively. These results clearly demonstrate the superiority of the J-ICP algorithm in pose estimation, not only enhancing the accuracy of attitude measurement but also significantly reducing system errors. This establishes a solid theoretical and experimental foundation for the practical application of pose estimation for non-cooperative spacecraft. In addition, we compared the computational effort of different algorithms and used computation time as an evaluation metric. In order to ensure that no other irrelevant variables were introduced in the process of fine alignment, all methods underwent a standardized preprocessing step on the initial input point cloud before analysis.