Formation control and coordinated tracking via asymptotic decoupling for Lagrangian multi-agent systems

We study the problem of formation control and trajectory tracking for a group of robotic systems modeled by Lagrangian dynamics. The objective is to achieve and maintain a stable formation for a group of multi-agent systems, while guaranteeing tracking of a specified trajectory. In order to do so, we partition the state space for the collective system into coordinates of the geometric center of mass of the group and coordinates that describe the relative positions of the robots with respect to the center of mass, thus defining the formation shape. The relative positions can be further partitioned in coordinates which describe the absolute distances and orientation of each robot to the center of mass. We can rewrite the total system as dynamics of the center of mass of the formation, and dynamics of the shape, where the systems are, in general, coupled. By imposing holonomic constraints between the subsystems (i.e.; imposing a configuration constraint) and hence reducing the system's dimension, we guarantee that the group can be driven to follow a desired trajectory as a unique rigid body. Using high gain feedback, we achieve asymptotic decoupling between the center of mass and the shape dynamics and the analysis is performed using a singular perturbation method. In fact, the resulting system is a singularly perturbed system where the shape dynamics describe the boundary layer while the center of mass dynamics describes the reduced system. After an initial fast transient in which the robots lock to the desired shape, a slower tracking phase follows in which the center of mass converges to a desired trajectory while maintaining a stable formation.