This paper describes a general geometric method for planar formations of non-holonomic systems. The approach directly provides the feasible controls that each individual robot has to execute in order for the team to maintain the formation based on the controls of a reference agent, either a real leader-robot or a virtual one. In order to directly satisfy the non-holonomic constraints, the geometric reasoning takes place in curvilinear coordinates, defined by the curvature of the reference trajectory, instead of the typical rectilinear coordinates. The generality of the approach lies on the ability to define dynamic formations so as to smoothly switch between static ones, where the robots can change both of their relative coordinates as they move, and the ability to acquire a desired formation given an initial random configuration. Furthermore, it is possible to correct errors in the achieved configuration of the vehicles on the fly. Simulated experiments are presented to verify the correctness of the provided derivations.
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