Robot swarms navigating through unknown obstacle environments are an emerging research area that faces challenges. Performing tasks in such environments requires swarms to achieve autonomous localization, perception, decision-making, control, and planning. The limited computational resources of onboard platforms present significant challenges for planning and control. Reactive planners offer low computational demands and high re-planning frequencies but lack predictive capabilities, often resulting in local minima. Multi-step planners can make multi-step predictions to reduce deadlocks, but they require substantial computation, resulting in a lower replanning frequency. This paper proposes a novel homotopic trajectory planning framework for a robot swarm that combines centralized homotopic trajectory planning (optimal virtual tube planning) with distributed control, enabling low-computation, high-frequency replanning, thereby uniting the strengths of multi-step and reactive planners. Based on multi-parametric programming, homotopic optimal trajectories are approximated by affine functions. The resulting approximate solutions have computational complexity $O(n_t)$, where $n_t$ is the number of trajectory parameters. This low complexity makes centralized planning of a large number of optimal trajectories practical and, when combined with distributed control, enables rapid, low-cost replanning.} The effectiveness of the proposed method is validated through several simulations and experiments.

翻译：机器人群体在未知障碍物环境中的导航是一个新兴研究领域，面临着诸多挑战。在此类环境中执行任务需要群体实现自主定位、感知、决策、控制与规划。机载平台有限的计算资源给规划与控制带来了显著困难。反应式规划器计算需求低、重规划频率高，但缺乏预测能力，常导致局部极小值问题。多步规划器可通过多步预测减少死锁，但需要大量计算，导致重规划频率较低。本文提出了一种新颖的机器人群体同伦轨迹规划框架，将集中式同伦轨迹规划（最优虚拟管道规划）与分布式控制相结合，实现了低计算量、高频率的重规划，从而融合了多步规划器和反应式规划器的优势。基于多参数规划，同伦最优轨迹可通过仿射函数近似。所得近似解的计算复杂度为$O(n_t)$，其中$n_t$为轨迹参数数量。这一低复杂度使得大量最优轨迹的集中式规划变得可行，并结合分布式控制实现快速低成本的重新规划。通过多项仿真和实验验证了所提方法的有效性。