This paper addresses the general problem of controlling a large number of robots required to move as a group. We propose an abstraction based on the definition of a map from the configuration space Q of the robots to a lower dimensional manifold A, whose dimension is independent of the number of robots. In this paper, we focus on planar fully actuated robots. We require that the manifold has a product structure A = G x S, where G is a Lie group, which captures the position and orientation of the ensemble in the chosen world coordinate frame, and S is a shape manifold, which is an intrinsic characterization of the team describing the “shape” as the area spanned by the robots. We design decoupled controllers for the group and shape variables. We derive controllers for individual robots that guarantee the desired behavior on A. These controllers can be realized by feedback that depends only on the current state of the robot and the state of the manifold A. This has the practical advantage of reducing the communication and sensing that is required and limiting the complexity of individual robot controllers, even for large numbers of robots
