Multi-agent deployment under the leader displacement measurement : a PDE-based approach

We study the deployment of a first-order multiagent system over a desired smooth curve in 3D space. We assume that the agents have access to the local information of the desired curve and their displacements with respect to their closest neighbors, whereas in addition a leader is able to measure his absolute displacement with respect to the desired curve. In this paper we consider the case that the desired curve is a closed C^2 curve and we assume that the leader transmit his measurement to other agents through a communication network. We start the algorithm with displacement based formation control protocol. Connections from this ODE model to a PDE model (heat equation), which can be seen as a reduced model, are then established. The resulting closed loop system is modeled as a heat equation with delay (due to the communication). The boundary condition is periodic since the desired curve is closed. By choosing appropriate controller gains (the diffusion coefficient and the gain multiplying the leader state), we can achieve any desired decay rate provided the delay is small enough. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical example illustrates the efficiency of the method.