## 3 Methodology
### Archive Update
This study uses the following (16) equation for calculating the angle between particles:  
\[\delta_{ij}=\delta\big{(}\mathbf{x_{i}},\mathbf{x_{j}}\big{)}=\arcsin\frac{f(\mathbf{x_{i} })\cdot f\big{(}\mathbf{x_{j}}\big{)}}{|f\left(\mathbf{x_{i}}\right)|\cdot|f\left(\mathbf{x _{j}}\right)|} \tag{16}\]  
instead of (17):  
\[\delta_{ij}=\delta\big{(}\mathbf{x_{i}},\mathbf{x_{j}}\big{)}=\arccos\frac{f(\mathbf{x_{i} })\cdot f\big{(}\mathbf{x_{j}}\big{)}}{|f\left(\mathbf{x_{i}}\right)|\cdot|f\left(\mathbf{x _{j}}\right)|} \tag{17}\]  
The benefits of using Equation (16) (based on arcsin) are as follows:  
1. Intuitive geometric interpretation and visualization: The range of arcsinX is closer to actual movement angles, making it easier to interpret and visualize particle movement paths. Due to its smaller range, arcsinX can better describe angle changes in physical systems, preventing significant angle jumps and facilitating fine-tuning. Assume a satellite is orbiting along a circular path around the Earth. The angle \(\theta\) between the satellite and a point on the Earth's surface can be represented as \(\theta\) = \(arcsin(d/h)\), where \(d\) is the distance along the Earth's surface from the point directly beneath the satellite to the point of interest, and \(h\) is the altitude of the satellite. We compare it to a linear approximation of the actual movement angles to show that arcsinX provides a closer approximation. The linear approximation can be written as \(\theta\ \approx\ d/h\), which is only accurate for tiny angles (\(d<<h\)). When we expand \(\theta=arcsin(d/h)\) with the Taylor series, we can use the general form of the Taylor series \(arcsin(x)\) using Equation (18). \[arcsin(x)=x+\frac{x^{3}}{6}+\frac{3x^{5}}{40}+O\Big{(}x^{7}\Big{)} \tag{18}\]  
For \(\theta=arcsin\left(\frac{d}{h}\right)\), we consider \(\frac{d}{h}\) as \(x\). Thus, the expansion becomes \(\theta=arcsin\left(\frac{d}{h}\right)=\frac{d}{h}+\frac{1}{6}\left(\frac{d}{h} \right)^{3}+\frac{3}{40}\left(\frac{d}{h}\right)^{5}+O\bigg{(}\left(\frac{d}{ h}\right)^{7}\bigg{)}\). This expansion shows that for small values of \(\frac{d}{h}\), the linear approximation \(\frac{d}{h}\) is reasonable. However, as \(\frac{d}{h}\) increases, higher-order terms become significant, making the approximation of \(arcsin\left(\frac{d}{h}\right)\) more accurate. Figure 4 illustrates a comparison between the calculated actual movement angles (\(\theta\)) using the \(arcsin(d/h)\) function and a straightforward linear approximation \((d/h)\). The plot demonstrates that the arcsin function (blue curve) closely follows the accurate movement angles, especially for larger values of d, illustrating its superior accuracy. In contrast, the linear approximation (red dashed line) deviates more as d increases, highlighting the effectiveness of the arcsin approach in accurately representing movement angles. 1. Numerical stability: \(arcsinX\) varies more smoothly within its range, and its derivative does not tend to infinity, resulting in more stable numerical computations. In contrast, \(arccosX\) has larger derivative changes near boundary values, which can cause numerical instability. 2. Fine-grained search: For multi-objective optimization problems in satellite data transmission, optimizing transmission paths and resource allocation requires high precision and fine adjustments. \(arcsinX\) offers finer search granularity, ensuring better solutionseven under complex constraints. The smooth and gradual changes of \(arcsinX\) lead to more uniform particle distribution and stable computations. Figure 4: Actual and approximate angles of movement in a simple linear approximation. In summary, Equation (16) based on \(arcsinX\) is more suitable for PSO algorithms dealing with complex multi-objective optimization problems, such as satellite data transmission systems, due to its intuitive geometric interpretation, numerical stability, and fine-grained search capability. 3.