Optimal leader allocation in UAV formation pairs ensuring cooperation

This paper considers the problem of optimally allocating the leader task between pairs of selfish unmanned aerial vehicles (UAVs) flying in formation. The UAV that follows the other achieves a fuel benefit. The noncooperative nature of the agents makes it necessary to arbitrate leader-allocation mechanisms that induce collaboration so that the fuel consumption benefits of flying in formation can be realized. Formulated as a nonlinear program, our problem poses two distinct challenges: on the one hand, given a fixed number of leader switches, determine the optimal leader allocation and, on the other, find the optimal number of leader switches. Even though the first problem is nonconvex, we identify a suitable restriction of its feasible set that makes it convex while maintaining the same optimal value. Regarding the second problem, our analysis of the optimal value of the problem as a function of the number of switches allows us to design a search algorithm which is guaranteed to find the solution in logarithmic time. Several simulations illustrate our results.