Mathematical Problems in Engineering

Volume 2015, Article ID 878536, 14 pages

http://dx.doi.org/10.1155/2015/878536

## Cooperative Solution of Multi-UAV Rendezvous Problem with Network Restrictions

Aviation University of Air Force, Changchun 130022, China

Received 2 February 2015; Accepted 11 May 2015

Academic Editor: Yang Tang

Copyright © 2015 Qingjie Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By considering the complex networks, the cooperative game based optimal consensus (CGOC) algorithm is proposed to solve the multi-UAV rendezvous problem in the mission area. Firstly, the mathematical description of the rendezvous problem is established, and the solving framework is provided based on the coordination variables and coordination function. It can decrease the transmission of the redundant information and reduce the influence of the limited network on the task. Secondly, the CGOC algorithm is presented for the UAVs in distributed cooperative manner, which can minimize the overall cost of the multi-UAV system. The CGOC control problem and the corresponding solving protocol are given by using the cooperative game theory and sensitivity parameter method. Then, the CGOC method of multi-UAV rendezvous problem is proposed, which focuses on the trajectory control of the platform rather than the path planning. Simulation results are given to demonstrate the effectiveness of the proposed CGOC method under complex network conditions and the benefit on the overall optimality and dynamic response.

#### 1. Introduction

In order to execute the missions such as simultaneous strike, cooperative reconnaissance, or SEAD (suppression of enemy air defense), multiple UAVs need to arrive at a selected region from different directions. Each UAV plans its path dynamically considering the restriction including enemy radars, missiles, and its own performance. For the sake of the successful task achievement, it requires that all UAVs arrive at the goal position simultaneously or sequentially, which is called rendezvous problem.

In respect to the time coordination problem, the method using the coordination function and coordination variables is adopted by [1–4]. The optimal flight control sequence of the UAVs can be obtained to meet certain performance index by this method. It can not only simplify the complexity of solving the problem but also reduce the redundant communication links among the UAVs. But it is still a kind of centralized control method in the view of the command and control manner. After the UAVs flying into the mission area, the communication will be reduced or interrupted. In order to ensure the survivability and increase the probability of the successful task, a distributed control method based on multiagent average consensus algorithm is proposed by [5–9]. It can deal with the case of battlefield environment changes. Due to the adoption of the average consensus algorithm, it is a “compromise” collaborative result. Therefore, the consensus-based flight control sequence is not considered in an optimal manner. To improve the performance of consensus algorithm, literature [10] proposes a noncooperative based optimal consensus (NCOC) algorithm, which can be employed to solve this rendezvous problem. However, the overall cost of multi-UAV system is usually not the minimum, since these UAVs only minimize their own cost function rather than the overall one. This is due to the fact that these UAVs are selfish or noncooperative. Literature [11, 12] focuses on the dynamic response and optimal cost by introducing the outdated state difference and the optimized weighted matrix to the consensus algorithm. This method accelerates the speed of the consensus convergence. Similarly, the consensus algorithm with virtual leader and state predictor has been adopted in [13] to increase the convergence speed of the rendezvous problem.

Different from the existing methods using average consensus algorithm [5–9], this paper focuses on the optimal solution rather than the convergence speed [11–13]. Further, the NCOC method [10] is developed, and the cooperative method of solving the rendezvous problem is proposed. The system achieves the overall optimal cost in this process. The novel method can deal with the complex network restriction such as switching topologies, communication delay.

The remainder of this paper is organized as follows. Firstly, the mathematical description of rendezvous problem is given. Based on the results of [5–9], the solving framework is proposed using the coordination function and coordination variable. A novel distributed control method using the cooperative game based optimal consensus (CGOC) algorithm is proposed for the “cooperative” UAVs. Finally, numerical experiments and simulation results are illustrated to show the effectiveness and benefit of the proposed method.

#### 2. Problem Description

##### 2.1. Basic Assumptions and Physical Constraints

This paper focuses on the distributed control method of the rendezvous problem, rather than the flight control of the platform. Hence, we assume that(1)all UAVs are small size;(2)each UAV is equipped with the autopilot, which can track the waypoint automatically;(3)mission control and flight control can be decoupled, respectively. Thus, the flight control problem can be simplified. In order to show the physical characteristics of the platform, the flight constraints and related performance parameters are listed as follows: (i)the maximum and minimum speed , ;(ii)the minimum turning radius ;(iii)the maximum flight time .

##### 2.2. Incomplete Kinematic Model of UAV

Similar to [1–7, 9], the 2D kinematic model of the UAV is selected to study the rendezvous problem ()where denotes the position vector of the th UAV and is the heading angle. is the velocity and is the changing rate of the heading angular velocity. and satisfyThe parameters , , and are determined by the physical performance.

The autopilot of each UAV maintains the expected heading angle and velocity. Its mathematical model can be described by two first-order differential equationsThe variable or vector with the superscript signifies the reference instruction. Parameters and are the constant coefficients of the heading and velocity channel of the autopilot.

##### 2.3. Communication Model

Graph theory is used to describe the communication model among multi-UAV system; see, Figure 1. Let denote the relationship between multiple UAVs with the set of nodes , the set of edges , and adjacent matrix . The node indices belong to a finite index set . The edge can be depicted by , and the value of corresponds to the edge of the graph; that is, . The neighbors set of node is defined by .