Recently, Unmanned Aerial Vehicles (UAVs) have emerged as relay platforms to maintain the connectivity of ground mobile ad hoc networks (MANETs). However, when deploying UAVs, existing methods have not consider one situation that there are already some UAVs deployed in the field. In this paper, we study a problem jointing the motion control of existing UAVs and the deployment of new UAVs so that the number of new deployed UAVs to maintain the connectivity of ground MANETs can be minimized. We firstly formulate the problem as a Minimum Steiner Tree problem with Existing Mobile Steiner points under Edge Length Bound constraints (MST-EMSELB) and prove the NP completeness of this problem. Then we propose three Existing UAVs Aware (EUA) approximate algorithms for the MST-EMSELB problem: Deploy-Before-Movement (DBM), Move-Before-Deployment (MBD), and Deploy-Across-Movement (DAM) algorithms. Both DBM and MBD algorithm decouple the joint problem and solve the deployment and movement problem one after another, while DAM algorithm optimizes the deployment and motion control problem crosswise and solves these two problems simultaneously. Simulation results demonstrate that all EUA algorithms have better performance than non-EUA algorithm. The DAM algorithm has better performance in all scenarios than MBD and DBM ones. Compared with DBM algorithm, the DAM algorithm can reduce at most 70% of the new UAVs number.

Unmanned Aerial Vehicles (UAVs) have emerged as promising relay platforms to improve networking performance (such as connectivity and throughput) for ground mobile ad hoc networks (MANETs) [

A variety of efforts have been made to explore the benefits of using UAVs as communication relays for ground MANETs. Some work wants to optimize the deployment of UAVs to improve the connectivity of ground nodes. Chandrashekar et al. presented a method on deploying minimum number of UAVs to connect a disconnected MANET [

However, existing works on the deployment of UAVs have not considered a situation that some UAVs have already been deployed in the field. With the movement of ground MANETs, existing UAVs may fail to connect all ground nodes. New UAVs need to be supplied to maintain the connectivity of ground MANETs. In order to minimize the number of new added UAVs, both the movement of existing UAVs and the deployment of new UAVs need to be considered. This is a joint optimization problem that optimizes both the deployment and motion control of multiple UAVs.

We consider the usage of existing UAVs by moving them to proper positions so that the number of new UAVs that is needed can be reduced. The existing UAVs have a limited motion range that depends on the speed of UAVs. In order to support bidirectional communication between UAVs and ground nodes, we assume that UAVs have the same communication range as ground nodes. Figure

An example that illustrates the importance of existing UAVs in maintaining the connectivity of ground MANETs.

In order to maintain the connectivity of ground MANETs, methods that do not consider existing UAVs such as [

To reduce capital expenditure, users would try their best to reduce the number of newly deployed UAVs. In other words, they will exploit the existing UAVs instead of ignoring them. By moving existing UAVs to proper positions and using existing UAVs as relays, the connectivity of ground MANETs can be improved. Figure

In this paper, we study the joint optimization problem of deployment and motion control of multiple UAVs so that the number of new added UAVs can be minimized. We firstly formulate this problem as a Minimum Steiner Tree problem with Existing Mobile Steiner nodes under Edge Length Bound constraints (MST-EMSELB) and prove the NP completeness of the problem. Then we present non-Existing UAVs Aware (non-EUA) algorithm and propose three Existing UAVs Aware (EUA) polynomial time approximation algorithms: Deploy-Before-Movement (DBM), Move-Before-Deployment (MBD), and Deploy-Across-Movement (DAM). The first two algorithms decouple the joint problem into deployment problem of new UAVs and motion control problem of existing UAVs. DBM algorithm optimizes the deployment of new UAVs before movement of existing UAVs and the MBD algorithm solves the problem contrarily. DAM algorithm is a mixed algorithm that solves the movement and deployment problem crossly. Simulation experiments show that all EUA algorithms have better performance in terms of new UAVs’ number than non-EUA algorithm. DAM algorithm is always better than DBM and MBD algorithms and can improve the performance to 70% at most comparing with DBM algorithm.

The main contributions of this paper are as follows:

We investigate a new problem in maintaining the connectivity of ground MANETs using multiple UAVs, which joints the deployment problem of new added UAVs and motion control problem of existing UAVs. We demonstrate the significance of considering existing UAVs to reduce the number of new added UAVs.

We formulate the problem as a Minimum Steiner Tree problem with Existing Mobile Steiner nodes under Edge Length Bound constraints (MST-EMSELB) and prove the NP completeness of the problem. Then, we propose three polynomial time approximation algorithms.

We compare the performance of proposed algorithms with non-EUA algorithms in simulation environment and demonstrate the effectiveness of proposed algorithms.

The rest of this paper is organized as follows. Section

Related works lie in two research fields: static relay deployment problem and mobile relay motion control problem.

Static relay deployment problems have been widely studied in wireless sensor networks (WSN). In WSN, it is hard to recharge the sensors after deployment. Due to the unbalanced load of message routing, some sensors will run out of energy before others. Then network partition happens and the whole network may be not available even though some sensors are still alive.

To extend the lifetime of WSN, some researches propose to deploy some static relay nodes in the field. Lloyd and Xue studied the problem of deploying minimum number of relay nodes so that, for each pair of sensor nodes, there is a connecting path consisting of relay and/or sensor nodes [

Similar to static relays, mobile relays were also proposed to extend the lifetime of WSN. But mobile relays often have rich resources and are able to move in the field. Thus mobile relays are more flexible than static relays and can be reused in different positions.

For WSN, the main purpose of using mobile relays is to save energy consumption of sensors and to extend the lifetime of the network. Wang et al. first studied the performance of a large dense network with one mobile relay and proposed a joint mobility and routing algorithm which can yield a network lifetime close to the upper bound [

As airborne platform of mobile relay, UAVs have been introduced to WSN, MANET, and other kinds of ground networks to improve the connectivity or link capacity between ground nodes. Since UAVs have relatively high mobility, sensibility, and self-controllability, they are especially suitable for MANET that has dynamic topology.

Chandrashekar et al. considered the problem of providing full connectivity to disconnected ground MANET nodes by dynamically placing Unmanned Aerial Vehicles (UAVs) to act as relay nodes [

With the development of UAV’s manufacturing technology, the size of UAV becomes smaller and the price also becomes cheaper. Thus it is possible to use a team of UAVs to provide network connection for ground nodes or improve their link capacity. Zhan et al. investigated a communication system in which UAVs are used as relays between ground-based terminals and a network base station [

We assume a scenario that multiple UAVs are used to maintain the connectivity of ground MANETs and some UAVs have already been deployed in the field. However, due to the movement of ground nodes and limited communication range, existing UAVs are not able to connect all ground nodes. Thus we need to add new UAVs to maintain the full connectivity of MANETs. The system model is shown in Figure

System model.

We assume the UAVs used in this paper are small four-rotor UAVs. The four-rotor UAVs can stay in a constant position and fly directly up and down. It can also spin 360 degrees around itself with a zero radius. To simplify the system, we assume all UAVs fly in different altitude so that collision avoidance of UAVs needs not to be considered in this paper.

Since the four-rotor UAV is small and always uses battery as energy, the velocity of the UAVs is limited. Because ground nodes are continuously moving, the task of motion control of existing UAVs and deployment of new UAVs must be finished within a given deadline. This requirement is especially important for some military scenarios. So we assume in this paper the mobility of existing UAVs is constrained. They can move towards any direction. However, the distance between new positions and current positions must not be more than a constant length. And we call this constant length motion range in this paper.

Connectivity represents the communication capability between nodes in a network. Here, the link capacity is used to represent the connectivity. The meaning of maintaining the connectivity of ground MANETs is to keep the link capacity higher than a given threshold. When the link capacity between two nodes is higher than the threshold, these two nodes are connected. Otherwise, they are disconnected.

Link capacity is the upper bound of data rate when transmitting data. According to Shannon equation the link capacity can be computed using (

Signal noise ratio (SNR) is the ratio of receiving power of signal with power of noise signal. The SNR of node

According to (

In this paper, we define the connectivity between any two nodes as a binary variable. If the distance between two nodes is not more than a constant length, we assume these two nodes are connected. Otherwise, these two nodes are disconnected. Here we name the constant length communication range. The communication range of ground nodes is smaller than the communication range between two UAVs or between one UAV and one ground node.

In our definitions, we assume that all current positions of ground nodes and existing UAVs are known. We also assume there are no obstacles that affect the mobility of UAVs or transmissions. Our problem can be described as follows: given a set of ground nodes and a set of existing UAVs, we want to find new positions for existing UAVs and positions for new added UAVs to form tree spanning all ground nodes so that the number of new added UAVs is minimized. There are two constraints in this problem. One is the distance between new position and current position of each existing UAV that is not more than a given motion range. The other is that length of each edge in the tree is no more than a given communication range.

Since this problem is similar to Steiner Tree Problem with minimum number of Steiner points, we formulate this problem as a Minimum Steiner Tree problem with Existing Mobile Steiner points under Edge Length Bound constraints (MST-EMSELB). The Steiner points here stand for UAVs and the Edge Length Bound is the communication range. The formal definition of MST-EMSELB problem is shown as follows.

In this section, we will prove that the decision version of the MST-EMSELB problem is NP complete. The NP hardness of the problem is proved by a polynomial time reduction from the Minimum Steiner Tree problem with Edge Length Bound constraints (MST-ELB) problem which is proved to be NP complete [

Given a set

Given a set

There is a polynomial time reduction from MST-ELB problem to MST-EMSELB problem.

MST-EMSELB problem is NP complete.

Given the number of new added Steiner points and a topology which specifies the edges in the final tree, a bottleneck tree under this given topology and edge bound constraint can be computed in polynomial time [

As previously mentioned, the MST-EMSELB problem belongs to NP complete problem; thus we try to find polynomial time approximation algorithms for this problem. In this section, we firstly present existing methods and then propose three heuristic algorithms for MST-EMSELB problem.

Currently, there are no particular algorithms designed for MST-EMSELB problem. The most related problem of MST-EMSELB is MST-EMS problem. When we crystallize the MST-EMS problem in the scenarios of using UAVs to maintain the connectivity of MANETs, a specific problem of MST-EMS is using new UAVs to maintain the connectivity of MANETs without considering existing UAVs.

Lin and Xue presented a minimum spanning tree (MST) based heuristic algorithm for MST-EMS problem whose worst case approximation ratio is 4 [

Since no mobile Steiner points are considered, the MST heuristic algorithm cannot be directly used for MST-EMSELB problem. Here we just take the Lin and Xue methods as a comparative method. Since this method has not considered the existing UAVs, we call this method non-Existing UAVs Aware (non-EUA) algorithm. Non-EUA algorithm just computes minimum number of new UAVs that is needed to connect all ground nodes. None of the existing UAVs will be reused for connecting ground MANETs. So the number of needed new UAVs computed by non-EUA should be an upper bound of other Existing UAVs Aware algorithms. The non-EUA algorithm is shown in Algorithm

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Since this is a joint optimization problem and there are two variables that need to be optimized, one variable is the new position of existing UAVs and the other variable is the position of new added UAVs. So we decouple the MST-EMSELB problem into two subproblems: the movement control problem of existing UAVs and the deployment problem of new added UAVs. In order to optimize the joint problem, we solve these two subproblems one by one. The first algorithm we proposed is Deploy-Before-Movement (DBM) algorithm that firstly optimizes the deployment of new UAVs and then optimizes the movement control of existing UAVs.

The main idea of DBM algorithm is shown as follows. Firstly we use the non-EUA algorithm to generate candidate positions of new added UAVs without considering existing UAVs. Then we match existing UAVs with candidate positions of new added UAVs. A match between an existing UAV and a candidate position of a new added UAV means that the new added UAV will be replaced by the existing UAV by moving this existing UAV to the candidate position. Since the motion range of existing UAVs is limited, the number of matches is also constrained. Here, we use Hungary algorithm to find maximum matches so that the number of needed new UAVs can be minimized. The DBM algorithm is shown in Algorithm

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Although DBM algorithm can utilize the mobility of existing UAVs, the candidate position for moved existing UAVs is limited which is the position computed by non-EUA algorithm. Due to the motion range limitation, existing UAVs may not be able to move to these candidate positions. Figure

A worst case for DBM algorithm.

However, the best solution for this case is shown in Figure

As previously mentioned, we decouple the joint optimization problem into two subproblems: the deployment of new UAVs problem and motion control problem of existing UAVs. DBM algorithm firstly solves the deployment problem and then solves the motion control problem. However, due to the less optimal motion control, DBM algorithm encounters some worst cases that none of existing UAVs can be reused. So we reverse the solution and propose Move-Before-Deployment (MBD) algorithm that firstly solves the movement problem and then solves the deployment problem.

The main idea of MBD algorithm is as follows. Firstly, we use a heuristic function to generate new positions of existing UAVs

The heuristic function mentioned in previous paragraph to generate new positions of existing UAVs might affect the performance of the whole MBD algorithm. We find that the DBM algorithm will be a perfect heuristic function to generate new positions for existing UAVs if the motion constraints of UAVs are released. This is because when the motion constraints of existing UAVs are released, they can move to any positions. So all of existing UAVs can match any candidate positions of new UAVs, since these candidate positions are computed by non-EUA algorithm that can minimize the number of new UAVs to maintain the connectivity of ground MANETs without considering existing UAVs. Thus, DBM algorithm can find the best positions for existing UAVs. The MBD algorithm is shown in Algorithm

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In MBD algorithm, the existing UAVs cut process is an important and necessary process to minimize the number of new added UAVs, because when constructing the minimum spanning tree

The existing UAVs cut process is shown in Figure

Existing UAVs cut process and new UAVs adding process of MBD algorithm.

Although MBD algorithm can minimize the number of new added UAVs using the existing UAVs cut process, the new positions of existing UAVs may not be optimal. This is because the new position of existing UAVs in MBD algorithm is generated using DBM algorithm which is based on the release of motion range constraints. So, when the motion range of existing UAVs is constrained, the performance of MBD algorithm will be depressed.

Since the deployment of new UAVs and the motion control of existing UAVs affect each other, we think that if we can solve the deployment and motion control problem crosswise, then the solution of the joint problem may be optimal. So here we propose a Deploy-Across-Movement (DAM) algorithm that solves these two problems simultaneously.

The main idea of DAM algorithm is as follows. Firstly, we generate a complete graph

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The New-UAV-Chain method in DAM algorithm is just the non-EUA algorithm in two-ground node case. The Existing-UAV-Chain (EUC) method is an important part of DAM algorithm since it controls the movement of existing UAVs. The details of Existing-UAV-Chain method are shown in Algorithm

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(7) choose one position

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Figure

Three candidate positions in EUC method.

In the proposed algorithms, we do not consider the power consumption of UAVs. In fact, the battery power of each UAV is limited and the limited power will certainly affect the deployment and movement of UAVs. Here, the power consumption is modeled as a constant factor. Existing UAVs that have very low battery power should land and recharge; thus it will be excluded from the set of existing UAVs. So all existing UAVs have adequate battery power. The battery power of UAVs can also be modeled as dynamic factor; thus existing UAVs may have different battery power due to duration time of their task. Then the deployment and movement UAVs should consider the power cost and optimize the survival time of the whole network. However, that is beyond the focus of this paper and we may consider it in our future work.

A simulation environment was set up to test the performance of the proposed algorithm. Simulation parameters are shown in Table

Simulation parameter.

Parameter | Value |
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Field size | 1–100 km^{2} |

Number of ground nodes | 10–100 |

Number of existing UAVs | 2–20 |

Communication range of ground nodes | 50–500 m |

Communication range of UAVs | 100–1000 m |

Motion range | 10–100 m |

To demonstrate the effectiveness of proposed Existing UAVs Aware (EUA) algorithms, we compare the performance in terms of new added UAV numbers with non-EUA algorithm and the CBBA (Consensus-Based Bundle Algorithm) proposed by Ponda et al. [

We use a square as testing field. We set the number of ground nodes as 50, existing UAVs as 5, communication range as 500 m, and motion range as 50 m. In this scenario, we vary the edge of testing field from 1 km to 10 km by increments of 1 km and randomly generate 100 different topologies for both ground nodes and existing UAVs.

Figure

Comparison with varying field size.

In this scenario, we set the edge of test field as 5 km and keep other three parameters the same while varying the number of ground nodes from 10 to 100.

Figure

Comparison with varying numbers of ground nodes.

In this case, we set number of ground nodes as 50 and kept the field size, number of ground nodes, communication range, and motion range static. We vary the number of existing UAVs from 2 to 20, by increments of 2.

Figure

Comparison with varying numbers of existing UAVs.

In this case, we set existing UAVs as 5 and keep the field size, number of ground nodes, number of existing UAVs, and communication range static. We vary the motion range from 10 m to 100 m, by increments of 10 m, and generate 100 different topologies for each motion range.

Figure

Comparison with varying motion range of existing UAVs.

In this case, we set motion range as 50 m while keeping the number of ground nodes, number of existing UAVs, and motion range static and vary the communication range of ground nodes from 50 m to 100 m, by increments of 50 m. The communication range between ground nodes and UAVs is set as twice communication range of ground nodes.

Figure

Comparison with varying communication range.

This paper studies the problem of using UAVs to maintain the connectivity of ground MANETs. Different from existing works, this paper considered a condition that some UAVs have already been deployed in the field. Due to the movement of ground MANETs and limited communication range, existing UAVs are not able to connect all ground nodes, so new UAVs need to be deployed to maintain the connectivity.

We present a joint optimization problem that combines the motion control of existing UAVs and the deployment of new added UAVs. We formulate this problem as a Minimum Steiner Tree problem with Existing Mobile Steiner points under Edge Length Bound constraints (MST-EMSELB) and prove NP completeness of the problem. We also propose three polynomial time heuristic algorithms named Deploy-Before-Movement (DBM), Move-Before-Deployment (MBD), and Deploy-Across-Movement (DAM) algorithms for the MST-EMSELB problem.

We demonstrate the effectiveness of proposed algorithms by comparing with non-Existing UAVs Aware (non-EUA) algorithm in simulation tests. We generate different scenarios by varying simulation parameters (including the number of ground nodes, number of existing UAVs, communication range, and motion range) and test the performance of both EUA and non-EUA algorithms. Simulation results show that EUA algorithms always have better performance than non-EUA method in terms of new added UAVs number. Among three EUA algorithms, MBD algorithm is better than DBM algorithm in most scenarios and DAM algorithm always has the best performance in all scenarios. In some scenarios, DAM algorithm can reduce at most 70% of new UAVs number compared with DBM algorithm.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the National Natural Science Foundation of China under Grant nos. 61379144, 61379145, and 61272485 for their supports of this work.