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 realtime 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.
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 [
The problem of cooperative search with multiple UAVs has been studied extensively due to its critical importance for a myriad of applications [
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 [
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. [
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 realtime updated probability map and uncertainty map, which can balance target search and environment exploration. Second, by extending the diffeomorphism [
The remainder of this paper is organized as follows. In Section
Let
Let
A connected partition
Given any finite set
Let
In this paper, the search environment is denoted by
Considering
Each UAV is equipped with a camera to take measurements within its Field of View (FoV). Supposing UAV
FoV of the UAV.
The search region
In this paper, the communication range is assumed to be limited. Let
The goal of this paper is to develop an efficient cooperative search method for multiple UAVs to find several unknown targets in a nonconvex environment with arbitrary obstacles. Inspired by the coverage control, an asynchronous distributed cooperative search framework is proposed by integrating the information update into the coverage control scheme, as shown in Figure
Asynchronous distributed cooperative search framework.
First, the cooperative search method in the convex region is presented, which is fundamental to that in the nonconvex region. At each decision, the UAV should determine the optimal control input based on the current state and environment information. The objective is to find several unknown targets, while maximizing the detection probability and minimizing the expected search time. To this end, the following objective function is proposed to be minimized, which is associated with the Voronoi partition:
In order to optimize the objective function
For simplification, the dynamics of each UAV is chosen as
However, in the nonconvex environment, the above control law may fail for two reasons. The centroid of Voronoi region may lie in the unfeasible area, making it impossible to converge to the centroid. The other is that the trajectory generated by the control law (
The density function represents the importance level of each cell in the region, which essentially affects the control strategy of UAVs in the search process. In this paper, we propose an adaptive density function
Here, the concept of the uncertainty map
From the above equations, it can be seen that if the target existence probability is close to the middle section of
This adaptive density function can promote the detection performance of target search while taking account of coverage performance. The probability map is used to lead UAV to move towards the region with high target existence probability, in order to find the targets in a shorter time. However, if the UAV has detected all the regions with high target probability and the centroid of Voronoi region no longer updates, the control law in (
In the cooperative search problem, each UAV keeps an individual probability map
Considering the uncertainty of sensor measurements, the probability map local update is related with whether a target is detected or not, provided that the target existence probability
(1) If a target is detected by UAV
If the target is in the FoV of UAV, the detection fault of missing the target occurs in one cell with the probability
Combining (
Based on the Bayesian complete probability formula,
Therefore, when the target is detected by UAV
(2) If the target is not detected by UAV
With the same evolving process as mentioned above, it can be obtained that
As described in (
Due to the “neighbor communication strategy,” only when UAV
Considering the search environment
Our transformation is inspired by the diffeomorphism idea [
Given two manifolds
For a given set
The above lemma follows that
As the main preprocessing procedure for constructing the transformation, the global convex decomposition method decomposes the original nonconvex region and the concave obstacles, making it possible to find the diffeomorphism based transformation in the nonconvex environment with concave obstacles.
First, the convex decomposition of concave obstacles is given. Some works have been conducted on the convex decomposition of concave polygons. Schachter [
Visible points from
The visible point, from which the decomposing line has the minimum weight value, is chosen. For each polygon, repeat the above steps until all the points of the decomposed polygons are convex.
Further, according to the decomposing lines and the edges of the decomposed obstacles, the whole search region is divided into a connected partition
First, extend the decomposing lines from two endpoints to intersect with region boundaries, defined as the dividing lines (for the whole region decomposition), while giving up extending those endpoints that will lead to intersecting with obstacles or other extending lines. Then, from each of those abandoned endpoints, draw a radial line along the two edges of the original concave obstacle to intersect with the region boundary or the obstacle boundary, and obtain the dividing lines. By these dividing lines, the whole region is divided into several subregions, each of which containing no more than one decomposed obstacle. Note that the feasible region of a new subregion may be separated into two parts by the decomposed obstacle. In this case, the subregion is divided again to generate a new small subregion without obstacles and the larger one with the obstacle.
Through convex decomposition, the whole region is divided into several subregions with no more than one convex obstacle. The method for constructing the diffeomorphism in Lemma
Let
Given
The situation is categorized according to the number of the common edges of the subregion and the obstacle.
(i) For the first case in Figure
Note that, through the diffeomorphism
(ii) For the second case in Figure
As claimed in the above six situations, it is obvious that the diffeomorphism
Diffeomorphism for the new decomposed regions.
The subregion containing no obstacle will be transformed by an identical transformation, which is also a diffeomorphism.
Let
Given the search region and obstacles, the diffeomorphism can be computed offline and stored in a table. Therefore, in the search procedure, the diffeomorphism can be directly used by looking up the table, which will significantly reduce the computation time.
Two examples of applying the convex decomposition method to divide the nonconvex region with arbitrary obstacles are shown in Figure
Convex decomposition.
Nonconvex region
Nonconvex region
Further, the diffeomorphism is constructed for each example region by using the method in Proposition
Diffeomorphism of the nonconvex region. (a) and (c) are the two examples of the original nonconvex region with 10 randomly sampled points. (b) and (d) are the transformed convex regions with the transformed image of the sampled points by diffeomorphism.
Sampled points in original
Point images in transformed
Sampled points in original
Point images in transformed
The diffeomorphism that transforms a complex nonconvex region with arbitrary obstacles to a convex region has been constructed. Now we will propose the cooperative search method in the nonconvex environment based on the transformation.
The objective function (
It can be seen that the allowable search domain is transformed from the feasible region
Since
So the gradient descent approach as (
Since the diffeomorphism
The following is an analysis of the convergence of the control law (
In the nonconvex region, the trajectories of UAVs governed by the control law (
Since the diffeomorphism
Assume
The following can be observed:
Thus, by LaSalle’s invariance principle, the trajectories
Therefore, the discretetime motion model and the real control law for UAV
Further, considering the mobility of the UAV, the control law must meet some constraints such that the generated trajectories are kinematically feasible:
(1) Maximum velocity: the maximum velocity
(2) Maximum turning angle: the maximum turning angle during a unit time period
Combining (
The cooperative search algorithm for the nonconvex region with arbitrary obstacles is presented as Algorithm
Cooperative search algorithm in the nonconvex environment with arbitrary obstacles is as follows:
Initially, all the UAVs are randomly deployed in the feasible region. The probability map and uncertainty map of each UAV are initialized.
The nonconvex region with arbitrary obstacles is divided by the global decomposition method, and the diffeomorphism
Through the diffeomorphism
The Voronoi partition of transformed region
At each time step, UAV
Update the probability map
Update the Voronoi partitions based on the new virtual positions of UAVs. Compute the generalized Voronoi centroids
If UAV
If the terminal condition is satisfied, the algorithm ends; otherwise, go to
In this section, some experiments are performed to verify the effectiveness of the cooperative search algorithm in different scenarios, and then the comparison with other algorithms is conducted to demonstrate the advantage of our algorithm in addressing the cooperative search problem in a nonconvex environment.
The algorithms have been programmed in Matlab. The simulation scenario is set as that five UAVs search for targets in two typical nonconvex regions with differently shaped obstacles. The whole search region
Simulation parameters.
UAV  Sensor  Detection model  

Maximum speed (m/s)  100 

45°  Positive sensing probability 
0.9 
Maximum turning angle  60° 

30°  False alarm probability 
0.1 
Flight height (m)  100 

30°  Target existence criteria 
0.9 
The first obstacle area is a typical concave region, represented as a red square. The convex decomposition and the diffeomorphism have been implemented as shown in Figures
Search flight trajectories of UAVs with time: (a–c) in the transformed region and (d–f) in the original region.
0 s
30 s
60 s
0 s
30 s
60 s
Probability map update for Scenario 1.
0 s
30 s
60 s
The nonconvex region in Scenario 2 contains two obstacles. The results of cooperative search are shown in Figures
Search flight trajectories of UAVs with time: (a–c) in the transformed region and (d–f) in the original region.
0 s
30 s
60 s
0 s
30 s
60 s
Probability map update for Scenario 2.
0 s
30 s
60 s
In order to verify the performance, the proposed algorithm is compared with a sweepline search algorithm [
We use the same simulation environment in Scenario 1. The above three algorithms are implemented and compared from the following aspects: average time required to find all the 6 targets, success rate of finding all the targets in a given time period of 500 s, and repeated detection rate. All the data are averaged by
Results of different search algorithms.
Sweepline  Random  Our cooperative  

search algorithm  search algorithm  search algorithm  
Average time required to search out all the targets (s)  461  N/A  56 
Success rate in 500 s  64%  2%  94% 
Repeated detection rate in 500 s  4.64  17.74%  6.45% 
From Table
Further, 20 targets are randomly deployed in the nonconvex region. The number of targets found with time evolving is used as the measure of performance, as shown in Figure
Comparison with sweepline search and random search algorithm.
Sweepline algorithm
Random search algorithm
Our cooperative algorithm
Number of targets found with time evolving
From all the results, it can be seen that our proposed cooperative search algorithm is quite efficient, for the search is led by the control law to optimize the information related objective function. And it can not only lead UAVs to converge to the regions with high target existence probability but also improve the exploration of the uncertain regions. Besides, the nonconvex environment is transformed into a convex one through the diffeomorphism at the beginning, so there is no excess step to deal with obstacle avoidance during the search process, dramatically speeding up the convergence.
This paper presents a distributed cooperative search method with multiple UAVs in a complex nonconvex environment with arbitrary obstacles. The proposed method mainly focuses on two issues: how to integrate the information update into the coverage control scheme and how to address the search problem in a complex nonconvex environment. First, an asynchronous distributed cooperative search framework is proposed, integrating information update into the coverage control scheme. The information update includes the Bayesian theory based individual update and a consensus fusion protocol based fusion update, while taking the inaccuracies and uncertainties in the sensor information into consideration. And an adaptive density function of coverage optimization is formulated based on the probability and uncertainty maps. Second, a new transformation method is proposed in order to extend the diffeomorphism to deal with the nonconvex environment with concave obstacles. The control strategy is presented considering the kinematic constraints, and the convergence is proved by LaSalle’s invariance principle. Finally, the effectiveness of the proposed algorithm is demonstrated through simulations. In future work, we will extend the algorithm to a highdimension space and take the complex dynamics of the UAV into consideration.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study was supported by the grant from the National Natural Science Foundation of China (61403406 and 61403410) and supported by the research project of National University of Defense Technology.