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.

翻译：随着机器人集群技术的快速发展和多样化应用，在复杂环境中导航机器人集群已成为一个关键研究方向。为确保安全导航并避免与障碍物发生潜在碰撞，虚拟管道的概念被引入以定义安全可航行区域。然而，当前虚拟管道内的控制方法面临拥塞问题，尤其在吞吐量较低的狭窄虚拟管道中。为应对这些挑战，我们首先创新性地引入虚拟管道面积与流通能力的概念，并建立了空间密度函数的新演化模型。其次，我们提出一种新颖的控制方法，该方法结合了用于集群导航的改进人工势场（APF）与用于分布调节的密度反馈控制，在此基础上设计了饱和速度指令。随后，我们生成了一个全局速度场，该速度场不仅能确保在虚拟管道内实现无碰撞导航，还能实现密度跟踪误差的局部输入到状态稳定性（LISS），这两点均得到了严格证明。最后，数值仿真与实际应用验证了所提方法在狭窄虚拟管道内管理机器人集群的有效性与优势。