An Efficient Real-Time Planning Method for Swarm Robotics Based on an Optimal Virtual Tube

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.