With the rapid development of robot swarm technology and its diverse applications, navigating robot swarms through complex environments has emerged as a critical research direction. To ensure safe navigation and avoid potential collisions with obstacles, the concept of virtual tubes has been introduced to define safe and navigable regions. However, current control methods in virtual tubes face the congestion issues, particularly in narrow virtual tubes with low throughput. To address these challenges, we first originally introduce the concepts of virtual tube area and flow capacity, and develop an new evolution model for the spatial density function. Next, we propose a novel control method that combines a modified artificial potential field (APF) for swarm navigation and density feedback control for distribution regulation, under which a saturated velocity command is designed. Then, we generate a global velocity field that not only ensures collision-free navigation through the virtual tube, but also achieves locally input-to-state stability (LISS) for density tracking errors, both of which are rigorously proven. Finally, numerical simulations and realistic applications validate the effectiveness and advantages of the proposed method in managing robot swarms within narrow virtual tubes.
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