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.

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