In recent years, there has been an increasing number of contributions dealing with the control of multiple agent formations. Of the many control strategies for formation control, distance-constrained formation control aims at maintaining inter-agent distances and utilizes relative measurements (i.e., distances and relative-positions) to generate the control action. In this work, we consider a collection of agents tasked with maintaining a distance constrained formation, where one agent in the ensemble is also designated as a leader and is subjected to an external velocity reference command.
The controller can act as the velocity input or the acceleration input depending on the mathematical model. We begin with a simple single integrator model without velocity reference by deriving the formation error dynamics and showing stability. By adding a velocity reference command to the leader and in the absence of any additional control action, the standard rigidity based formation stabilization solutions will exhibit a steady state formation error. As we try to address this problem we also reveal more interesting relations between the upper bound of the steady state error and the graph properties. Our approach is to combine gradient based formation controller with a proportional and integral (PI) control to eliminate this steady-state error. We show that such a scheme preserves the stability properties of the formation error dynamics as well as properties of the networked system’s centroid. We then explore the properties of double integrator model and show that using rigidity based formation controller only is not enough to ensure stability. By adding a velocity-based decentralized control we were able to reach velocity consensus and assure formation stability. As we want the system to also follow an external reference velocity, a simple inner-loop controller is implemented on the leader to ensure velocity tracking. Numerical simulations are shown to illustrate the theoretical results.
