## 5 Conclusions  
We have proposed a minimum angle particle swarm optimization (MAPSO) algorithm to solve multi-objective satellite data transmission scheduling problems. The proposed algorithm was encoded as a discrete optimizer, and the calculation equation of the sine function was modified to enhance the PSO algorithm's ability to deal with complex multi-objective problems. The MAPSO method introduces this new minimum angle selection strategy into the design and effectively selects the global optimal solution by calculating the minimum angle between particles. MAPSO balances convergence and diversity by avoiding local extrema in the global search process. In the experimental validation section, we tested the MAPSO method on 24-scale task instances, demonstrating its ability to solve multi-objective data transmission scheduling problems and improve data transmission efficiency and accuracy. The experimental results show that the MAPSO method has an advantage in solving data transmission problems and maintains high solution quality across different task scales, indicating its generalization capability and adaptability in practical applications. Future research can further optimize this method, combining it with other intelligent optimization techniques to enhance its application potential in more extensive and complex task environments, promoting the development of satellite communication technology. Data curation, YS., Z.W. and H.R. ; writing--original draft, Z.Z. ; writing--review and editing, S.C. and L.X. ; supervision, L.X. All authors have read and agreed to the published version of the manuscript. This work is partially supported by the National Natural Science Foundation of China (Grant No. 61806119), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2024 JC-YBMS-516), and Fundamental Research Funds for the Central Universities (No. GK20220104). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Experiment** & **MAPSO** & **DRSC-MOAGDE** & **DSC-MOPSO** & **MO-Ring-PSO-SCD** & **NSGA-II** & **MOEA/D** & **MOGA** & \(\left(\textbf{r}_{1},\textbf{r}_{2}\right)\) \\ \hline c5 & 1.8643 & 1.933 & 1.8781 & 1.759 & 1.4546 & 1.3109 & 0.4688 & (22,000, 350) \\ c6 & 2.1385 & 2.2142 & 2.164 & 2.0117 & 1.5687 & 1.4493 & 0.6678 & (40,000, 800) \\ c7 & 2.4821 & 2.6056 & 2.5382 & 2.3015 & 1.8887 & 1.6887 & 0.9002 & (62,000, 1100) \\ c8 & 2.5811 & 2.6837 & 2.6687 & 2.384 & 1.9333 & 1.7056 & 0.8436 & (68,000, 1500) \\ c9 & 2.8036 & 2.9306 & 2.9146 & 2.5939 & 2.0256 & 1.7963 & 1.0452 & (69,000, 1700) \\ c10 & 1.3491 & 1.4798 & 1.4504 & 1.1346 & 1.1349 & 0.8035 & 0.5479 & (23,000, 400) \\ c11 & 1.3913 & 1.4619 & 1.4554 & 1.1566 & 1.1587 & 0.8288 & 0.5787 & (40,000, 800) \\ c12 & 1.5034 & 1.4123 & 1.3879 & 1.2595 & 1.2201 & 0.9312 & 0.6761 & (54,000, 1200) \\ c13 & 1.6304 & 1.5168 & 1.4953 & 1.3692 & 1.3274 & 1.0559 & 0.7779 & (56,000, 1500) \\ c14 & 1.7058 & 1.5845 & 1.5612 & 1.4668 & 1.3804 & 1.1115 & 0.808 & (57,000, 2000) \\ c15 & 1.8316 & 1.7343 & 1.7267 & 1.5777 & 1.412 & 1.2063 & 0.9465 & (58,000, 2500) \\ c16 & 2.0992 & 1.9974 & 1.9621 & 1.85 & 1.5967 & 0.7781 & 0.412 & (14,000, 400) \\ c17 & 2.3119 & 2.1932 & 2.1501 & 2.0406 & 1.7374 & 0.8672 & 0.4618 & (30,000, 850) \\ c18 & 2.6322 & 2.5113 & 2.4778 & 2.3618 & 2.0116 & 1.1221 & 0.6498 & (40,000, 1200) \\ c19 & 2.9624 & 2.8154 & 2.829 & 2.6767 & 2.2743 & 1.3069 & 0.7946 & (50,000, 1500) \\ c20 & 3.4458 & 3.312 & 3.2611 & 3.1492 & 2.7236 & 1.6766 & 1.1481 & (53,000, 1600) \\ c21 & 3.6943 & 3.5292 & 3.4659 & 3.366 & 2.9313 & 1.7517 & 1.3172 & (55,000, 2100) \\ c22 & 3.9328 & 3.7508 & 3.7452 & 3.55 & 3.1034 & 1.7862 & 1.3576 & (60,000, 2500) \\ c23 & 4.0047 & 3.8169 & 3.8017 & 3.5916 & 3.1393 & 1.6375 & 1.2498 & (55,000, 2800) \\ c24 & 4.1859 & 4.0126 & 3.9654 & 3.7728 & 3.2642 & 1.8505 & 1.306 & (56,000, 3100) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Hypervolume value. Data Availability Statement: The data supporting the reported results can be found in the manuscript.