This paper addresses the design of simple robot behaviors that realize emergent group behaviors. We present a method to coordinate a large number of under-actuated robots by designing control laws on a small dimensional abstraction manifold, independent of the number and ordering of the robots. The abstraction manifold has a product structure consisting of elements of a Lie group that capture the position and orientation of the ensemble in the world frame, and elements of a shape manifold that provide an intrinsic description of the distribution of team members relative to one another. We design decoupled controls for regulating the group and the shape variables. The realization of the controller on each robot requires the feedback of the robot state, and the state on the abstraction manifold. We present experimental results with a team of five car-like robots equipped with omnidirectional cameras and IEEE 802.11b networking.
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