Beamforming and Trajectory Planning Method Under Fixed-Footprint Conditions for Multi-HAPS Systems

High Altitude Platform Station (HAPS) is a new airborne communications platform that provides wide-area communications services from the stratosphere and has the potential for coverage extension in 6G networks. In particular, it is a practical scenario where multiple HAPS work together to provide communications in densely populated urban areas and extensive regions. However, changes in ground coverage due to HAPS movement cause frequent handovers, which pose challenges to communication quality and stability. To solve this problem, “footprint fixation,” which maintains constant ground coverage even when the HAPS moves, is expected to reduce handovers and improve communication quality. How we should design beamforming and trajectory under fixed footprint conditions has not been clarified. In this paper, we propose a beamforming and trajectory planning method under fixed-footprint conditions for multiple HAPS systems. For footprint fixation, beamforming dynamically adapts to HAPS motion to eliminate ground coverage shifts, reduce handover frequency, and improve communication stability. Under fixed-footprint conditions, trajectory planning aims to improve the throughput of UEs with low (5th percentile) and medium (50th percentile) communication quality simultaneously through sequential multi-objective optimization. Simulation results in major Japanese cities with different UE distributions show that the proposed method improves both low and medium communication quality UEs, achieving a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{2 0 - 3 6 \%}$</tex> improvement in 5th percentile throughput and an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$8-20 {\%}$</tex> improvement in 50th percentile throughput compared to other methods. In addition, the footprint fixation increases coverage stability, reducing outage probability to less than 2 % and significantly reducing handover frequency.