To tackle the distributed power optimization problems in wireless sensor networks localization systems, we model the problem as a hierarchical game, i.e. a multi-leader multi-follower Stackelberg game. Existing researches focus on the power allocation of anchor nodes for ranging signals or the power management of agent nodes for cooperative localization, individually. However, the power optimizations for different nodes are indiscerptible due to the common objective of localization accuracy. So it is a new challenging task when the power allocation strategies are considered for anchor and agent nodes simultaneously. To cope with this problem, a hierarchical game is proposed where anchor nodes are modeled as leaders and agent nodes are modeled as followers. Then, we prove that games of leaders and followers are both potential games, which guarantees the Nash equilibrium (NE) of each game. Moreover, the existence of Stackelberg equilibrium (SE) is proved and achieved by the best response dynamics. Simulation results demonstrate that the proposed algorithm can have better localization accuracy compared with the decomposed algorithm and uniform strategy.

Positioning information plays a significant role in multi-unmanned aerial vehicles (UAVs) applications. Traditionally, the positioning information is widely provided by Global Navigation Satellite System (GNSS) due to its good performance and global coverage. However, owing to complicated flight environment or signal blockage, jamming and unintentional interference, the UAVs may fail to locate themselves by using GNSS alone. As a new method to resolve these problems, cooperative positioning, by incorporating peer-to-peer range measurements and assisted information, has attracted more and more attentions due to its ability to enhance the accuracy and availability of positioning. However, achieving good performance of cooperative positioning of multi-UAVs is challenging as their mobility, arbitrary nonlinear state-evolution, measurement models and limited computation and communication resources. In this paper, we present a factor graph (FG) representation and message passing methodology to solve cooperative positioning problem among UAVs in 3-dimensional environment where GNSS cannot provide services. Moreover, to deal with the nonlinear state-evolution and measurement models while decreasing the computation complexity and communication cost, we develop a distributed algorithm for dynamic and hybrid UAVs by means of Spherical-Radial Cubature Rules (CR) method with belief propagation (BP) and variational message passing (VMP) methods (CRBP-VMP) on the FG. The proposed CRBP deals with the highly non-linear state-evolution models and non-Gaussian distributions, the VMP method is employed for ranging message, gets the simpler message representation and can reduce communication cost in the joint estimation problem. Simulation results demonstrate that the higher positioning accuracy, the better convergence as well as low computational complexity and communication cost of the proposed CRBP-VMP algorithm, which can be achieved compared with sum-product algorithm over a wireless network (SPAWN) and traditional Cubature Kalman Filters (CKF) method.