Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 196730, 19 pages

http://dx.doi.org/10.1155/2015/196730

## Cooperative Search by Multiple Unmanned Aerial Vehicles in a Nonconvex Environment

College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, China

Received 12 January 2015; Revised 18 July 2015; Accepted 26 July 2015

Academic Editor: Alain Vande Wouwer

Copyright © 2015 Xiaoting Ji 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

This paper presents a distributed cooperative search algorithm for multiple unmanned aerial vehicles (UAVs) with limited sensing and communication capabilities in a nonconvex environment. The objective is to control multiple UAVs to find several unknown targets deployed in a given region, while minimizing the expected search time and avoiding obstacles. First, an asynchronous distributed cooperative search framework is proposed by integrating the information update into the coverage control scheme. And an adaptive density function is designed based on the real-time updated probability map and uncertainty map, which can balance target detection and environment exploration. Second, in order to handle nonconvex environment with arbitrary obstacles, a new transformation method is proposed to transform the cooperative search problem in the nonconvex region into an equivalent one in the convex region. Furthermore, a control strategy for cooperative search is proposed to plan feasible trajectories for UAVs under the kinematic constraints, and the convergence is proved by LaSalle’s invariance principle. Finally, by simulation results, it can be seen that our proposed algorithm is effective to handle the search problem in the nonconvex environment and efficient to find targets in shorter time compared with other algorithms.

#### 1. Introduction

Over the past decade, unmanned air vehicles (UAVs) with functional diversity and low cost have been extensively employed in many civil and military applications, such as environment surveillance, battle reconnaissance, and search and rescue in the hazardous environment [1–3]. With the development of advanced sensing and information processing technology, cooperative search has been one of the most popular utilizations of UAVs equipped with sensors (such as camera, Lidar, and sonar) [4]. The goal of cooperative search is to control multiple UAVs to find several unknown targets deployed in a given region, while maximizing the detection probability and minimizing the expected search time.

The problem of cooperative search with multiple UAVs has been studied extensively due to its critical importance for a myriad of applications [3, 5, 6]. Existing methods can be classified into two categories: predefined flight paths based search and dynamic path planning based search. The former is to generate the flight paths (e.g., parallel lines or outward spirals) in advance and follow these paths during search execution. The typical method of this category is sweep-line based search [7], in which the agents sweep all the points in the given area to find the targets. This method is effective so that no search areas are missed but not efficient due to the predefined paths and cannot be used for searching the dynamic targets. The latter method is to convert the cooperative search problem into a multiagent path planning issue. Typically, the path planning problem is formulated as the optimization of a team objective function subject to a set of constraints [8]. Therefore, dynamic programming [9], artificial intelligence [5, 6], and model predictive control (MPC) [10] can be used for solving these problems. In this paper, inspired by the coverage control, a distributed cooperative search method is proposed by integrating information update into the coverage control scheme, which also belongs to path planning based search methods. In recent years, with the development of related theory in Mobile Wireless Sensor Network (MWSN), coverage control methods [11] have been widely used in multiple robots system. The purpose of coverage control is to optimally deploy the mobile sensors to maximize the coverage of environment. Due to the limited Field of View (FoV), the cooperative search problem with multiple UAVs can be treated as an optimal coverage problem with a bounded sensing size of sensors [12]. And the distributed control strategy can be easily obtained by optimizing the objective function through the Lloyd algorithm [11]. Besides, by using the coverage control scheme in the search problem, it will not only promote the detection performance of target search but also take account of coverage performance. However, few of the existing coverage control methods have considered the information update about target existence probability, which is crucial to the search problem in practice and essentially affects the movements of UAVs.

Therefore, the first issue that should be addressed is how to integrate the information update into the coverage control scheme in order to solve the search problem. To our knowledge, there are only a few works that utilize the coverage control method with consideration of information update. Zhong and Cassandras [13] use coverage control and data collection to maximize the joint detection probability of random events in a given mission space. However, their goal is to find the optimal deployments for detection, which is essentially different from the search problem. And the density function (importance level of each cell in the region) in their objective formulation depends on the distance to the critical points (i.e., the detected sources). Mirzaei et al. [14] use two different types of vehicles for search and coverage tasks. Search vehicles use a limited look-ahead dynamic programming algorithm to maximize the amount of information gathered by the whole team, while service vehicles use the coverage control method to spread out over the environment to optimally cover the terrain. However, the search and coverage control are executed on different platforms by different schemes, and only the probability distribution of critical points is utilized in the objective function. Besides, the information fusion update is not considered. Hu et al. [15] formulate the path planning as a coverage control problem to find an optimal configuration of all agents that minimizes a given coverage performance cost function. They use the uncertainty information as the density function to cover the region uniformly, which cannot directly facilitate agents to search the region with high target existence. In this paper, we integrate the information update into the coverage control scheme by introducing the probability map of target existence together with the uncertainty map into the density function of coverage optimization. In most coverage control problems, the density function is fixed [16, 17] or only depends on the distance to some critical points [18, 19]. However, our new density function can be adaptively changed, depending on the real-time updated target existence probability and uncertainty map. The higher the probability map is, the larger the probability of target existence is. So utilizing the probability map as the density function can help UAV converge to the regions with high target existence and find targets in a short time. If all the regions with high target existence probability have been searched, or the UAV gets stuck in the local optimum, the uncertainty map can help the UAV escape from the local optimum and explore more regions.

Due to practical requirements, realistic search areas may be arbitrarily shaped with arbitrary obstacles. Thus, the other issue in our search problem is how to handle the nonconvex environment during search execution. Pimenta et al. [12] use the geodesic distance based algorithm to solve the coverage problem with a network of heterogeneous robots in the nonconvex environment. However, it may cause the robot to get stuck in a saddle point, or even worse, to drive into an obstacle, as the projections of the gradients into target direction of the geodesic distance do not add up to zero when reaching the boundary. Teraoka et al. [19] and Adibi et al. [20] present the potential field-based approaches representing control under the influence of virtual force generated by goals (attraction) and obstacles (repulsion), which have advantages in less computation and disadvantages in being easily trapped into a local minimum of the potential field. Breitenmoser et al. [18] present an algorithm which combines classical Voronoi coverage with Lloyd algorithm and the local planning algorithm TangentBug to compute the motion of the robots around obstacles and corners. It assumes that the range of sensor is infinite, meaning that the sensors can “see” through obstacles, which is unrealistic in practice. Caicedo et al. [16, 17] map a class of connected regions with holes to an almost convex region through a diffeomorphism, such that Voronoi partition and Lloyd algorithm [11] can be used to solve the coverage problem. However, the obstacles in their work must be simple convex ones, and the kinematic model of agents is not considered. As mentioned above, cooperative search in the nonconvex environment with arbitrary obstacles is still an open problem, especially when taking more realistic detection and the kinematic model into consideration. Motivated by this, we propose a generalized method to construct a transformation that can transform the nonconvex region with arbitrary obstacles into an almost convex region. Combining with the asynchronous distributed cooperative search framework, the cooperative search problem in the complex nonconvex environment can be solved.

The main contributions of this work are as follows. First, an asynchronous distributed cooperative search framework is developed that integrates information update into coverage control scheme. And an adaptive density function is formulated depending on real-time updated probability map and uncertainty map, which can balance target search and environment exploration. Second, by extending the diffeomorphism [16], we propose a new transformation method to handle the nonconvex environment with arbitrary obstacles, not limited to convex ones. Based on the transformation, the cooperative search algorithm is provided that can address search problem in the nonconvex environment with arbitrary obstacles. Finally, a control strategy is designed considering the kinematic constraints of UAV, and the convergence is proved using LaSalle’s invariance principle.

The remainder of this paper is organized as follows. In Section 2, some useful preliminaries are provided. The cooperative search framework and the objective formulation, as well as the explicit information update procedure, are presented in Section 3. In Section 4, the generalized method for constructing a transformation is presented to handle the nonconvex environment with arbitrary obstacles. Then the cooperative search algorithm in the nonconvex environment is proposed, and the stability and convergence are proved. Simulation results with analysis are shown in Section 5, and the conclusions are drawn in Section 6.

#### 2. Preliminaries

##### 2.1. Basic Notions and Definitions

Let be the set of real numbers and let denote the -dimensional Euclidean space. For a given set , and denote the interior and the boundary of , respectively. One has . represents the Euclidean distance between points , and is a closed ball centered at with radius . In the following, some definitions about the environment are presented.

*Definition 1 (connected partition [21]). *Let be a finite set and let be a subset of ; if , , and , is said to be a connected partition of .

*Definition 2 (Voronoi partition). *A connected partition of is said to be a Voronoi partition of , generated by a vector of distinct points , if for each there exists that satisfies , , .

*Definition 3 (nonconvex [22]). *Given any finite set , point of is nonconvex if, for all , there exist and in so that the open interval is outside .

*Definition 4 (nonconvex allowable environment [22]). *Let be a finite set that is allowable if (i)is compact and connected;(ii) is continuously differentiable except on a finite number of points;(iii) has a finite number of nonconvex points which are either isolated points or arcs continuously differentiable everywhere except, possibly, at their end points.

In this paper, the search environment is denoted by , which is a polygon in . It is assumed that there are several polygonal obstacles , within . Thus, the overall feasible region is . Obviously, the search region is nonconvex since it contains several obstacles.

##### 2.2. Detection and Communication Model

Considering UAVs in the search region, each UAV is modeled as a nonholonomic point mass moving at a limited speed with a minimum turning radius. The position of UAV is denoted by , and the heading angle is . So the state of UAV can be represented as .

Each UAV is equipped with a camera to take measurements within its Field of View (FoV). Supposing UAV is aiming at point at time (Figure 1), the formulation of FoV iswhere represents the flight height of UAV and , , and are the installed angle and view angles of the camera, related to the physical performance.