Multi-agent motion planning using Burgers' viscous equation and Hopf-Cole transformation, a flatness-based implementation

In a previous work, we showed how the flatness of the heat equation with two controls at both ends of the domain could be used to generate solutions to create solutions to Burgers' equation. The created solutions were thought as a motion planner for a multi-agent systems of e.g. ground robots, UAVs. The trajectories of the different particles of the PDE are to be used as input for a feed-forward control. In this paper, we show a practical implementation of this method, oriented toward computational efficiency based on Particles Swarm Optimization. In a first section we introduce the flatness based mathematical framework of our planner. We present the implementation in a second part. We present examples of trajectories in a last section and study and discuss the performances and the properties of the generated trajectories before drawing conclusions and future use of the framework in real UAVs systems.