Nash Bargaining Solution based rendezvous guidance of unmanned aerial vehicles

Abstract

This paper addresses a finite-time rendezvous problem for a group of unmanned aerial vehicles (UAVs), in the absence of a leader or a reference trajectory. When the UAVs do not cooperate, they are assumed to use Nash equilibrium strategies (NES). However, when the UAVs can communicate among themselves, they can implement cooperative game theoretic strategies for mutual benefit. In a convex linear quadratic differential game (LQDG), a Pareto-optimal solution (POS) is obtained when the UAVs jointly minimize a team cost functional, which is constructed through a convex combination of individual cost functionals. This paper proposes an algorithm to determine the convex combination of weights corresponding to the Pareto-optimal Nash Bargaining Solution (NBS), which offers each UAV a lower cost than that incurred from the NES. Conditions on the cost functions that make the proposed algorithm converge to the NBS are presented. A UAV, programmed to choose its strategies at a given time based upon cost-to-go estimates for the rest of the game duration, may switch to NES finding it to be more beneficial than continuing with a cooperative strategy it previously agreed upon with the other UAVs. For such scenarios, a renegotiation method, that makes use of the proposed algorithm to obtain the NBS corresponding to the state of the game at an intermediate time, is proposed. This renegotiation method helps to establish cooperation between UAVs and prevents non-cooperative behaviour. In this context, the conditions of time consistency of a cooperative solution have been derived in connection to LQDG. The efficacy of the guidance law derived from the proposed algorithm is illustrated through simulations.