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Motivated by recent research on cooperative unmanned aerial vehicles (UAVs), this paper introduces a new cooperative distributed trajectory optimization approach for systems with independent dynamics but coupled objectives and hard constraints. The overall goal is to develop a distributed approach that solves small subproblems while minimizing a fleet-level objective. In the new algorithm, vehicles solve their subproblems in sequence while generating feasible modifications to the prediction of other vehicles' plans. In order to avoid reproducing the global optimization, the decisions of other vehicles are parameterized using a much smaller number of variables than in the centralized formulation. This reduced number of variables is sufficient to improve the cooperation between vehicles without significantly increasing the computational effort involved. The resulting algorithm is shown to be robustly feasible under the action of unknown but bounded disturbances. Furthermore, the fleet objective function is proven to monotonically decrease as the algorithm cycles through the vehicles in the fleet and over the time. The results from simulations and a hardware experiment demonstrate that the proposed algorithm can improve the fleet objective by temporarily having one vehicle sacrifice its individual objective, showing the cooperative behavior.