A leader–follower formation control approach for target hunting by multiple autonomous underwater vehicle in three-dimensional underwater environments

Abstract

As one of the challenging tasks of multiple autonomous underwater vehicles systems, the realization of target hunting is the great significance. The multiple autonomous underwater vehicle target hunting is studied in this article. In some research, because the hunting members cannot reach the hunting point at the same time, the hunting time is long or the target escapes. To improve the efficiency of the target hunting, the leader–follower formation algorithm is introduced. Firstly, the task is assigned based on the distance between the autonomous underwater vehicle and the target. Then, the autonomous underwater vehicles with the same task are formed based on leader–follower mode, and the formation is kept to track the target. In the final capture phase, multiple autonomous underwater vehicle system use angle matching algorithm to round up target. The simulation results show that the proposed algorithm can effectively accomplish the target hunting task, save the hunting time, and avoid the target escape. Compared with the bioinspired neural network algorithm, the proposed algorithm shows better performance.

Introduction

As the complexity of underwater missions increases, it is difficult for a single autonomous underwater vehicle (AUV) to complete the task. ^1,2 Multi-AUV collaboration has become a hot topic of research. Target hunting is one of the challenging tasks in multi-AUV systems. ^3,4 There have been many studies on the target hunting of multi-robots and some results have been achieved. For example, Cao et al. ⁵ proposed a distributed control algorithm based on local collaboration to achieve multi-robot hunting of targets. The algorithm improves the hunting success rate through collaboration. Song et al. ⁶ established a steady-state mathematical model for multi-robot hunting behavior and improved the stability of target hunting. Denzinger and Fuchs ⁷ proposed a target hunting strategy that combines neighborhood rules and genetic algorithms. The algorithm classifies the current state with the nearest neighbor rule and then applies the genetic algorithm to optimize the hunting behavior. These algorithms are capable of achieving target hunting and can optimize the path of the hunting. However, they may cause a target hunting failure in changing environments.

Negotiation and calculation are common methods for resolving target hunting but can lead to delays in robotic action. ^{8 –10} When the environments change, robot spends more time replanning the path, so these methods are not suitable for target hunting in a dynamic environments. Yamaguchi ¹¹ studied the target hunting in dynamic environments based on Hilare-type mobile robot and solved the hunting problem using smooth time-varying feedback control method. This control strategy improves efficiency, but the robot may fall into a “dead zone” in a complex environments with a large number of obstacles.

To solve the impact of the environments on target hunting, the potential field function is constructed for the environments and embedded in the hunting algorithm to achieve target hunting in complex environments. For example, Wu et al. ¹² and Cao et al. ¹³ applied the artificial potential field and the limit cycle algorithm to the formation control to complete the cooperative hunting. Cao et al. ¹⁴ used the environmental potential field function to adjust the fitness function of the Particle Swarm Optimization (PSO) algorithm to adapt to the complex hunting environments. Rasekhipour et al. ¹⁵ proposed a predictive path controller based on potential field function to control multi-robots to achieve target hunting. The introduction of the environmental potential field improves the adaptability of the algorithm, but the efficiency of the hunting is still to be further improved.

Inspired by the good performance of bioinspired neural networks (BNN) in complex systems, Ni and Yang ¹⁶ applied BNN to real-time cooperative hunting in unknown environments. The results show that the proposed algorithm can guide the robots to effectively capture the targets and be able to handle the failure in the hunting process. However, the disadvantage of using the BNN planning path is that the computational complexity is high and the path planning time is long.

These studies are based on the target hunting problem of multiple mobile robots. The underwater hunting is more complicated than the ground hunting. Not only is the three-dimensional (3-D) environments, but each AUV has six degrees of freedom. Nguyen and Hopkin ¹⁷ and Williams ¹⁸ modeled AUV ore-seeking and applied a full coverage method to achieve hunting. However, the algorithm only studies the hunting of static targets without intelligence. Huang and Zhu ¹⁹ studied the multi-AUV target hunting problem based on BNN model. The algorithm uses distance-based negotiation for task assignments. Simulation experiments are carried out in 2-D and 3-D environments. The proposed algorithm can automatically handle various states and capture targets efficiently. However, the hunting fails because of the conflict. Later, Cao et al. ²⁰ proposed a local prediction method to resolve conflicts. AUVs participating in the hunting are predicting the behavior of other AUVs to avoid conflicts before they act. Compared with the artificial potential field algorithm, the algorithm reduces the navigation distance. However, in the literature, ^17–20 the hunting team was formed before the negotiations, and sometimes it is unreasonable. Chen and Zhu ²¹ proposed a hunting algorithm that combines Glasius bioinspired neural networks (GBNN) and reliability functions. The algorithm solves the hunting of heterogeneous multi-AUV pairs with intelligent targets in a dynamic multi-obstacle underwater environments. Ni et al. ²² divided the multi-AUV target hunting into two stages: search and capture. An improved spinal nervous system algorithm to control the search formation in the search phase. In the capture phase, a dynamic alliance method based on two-way negotiation strategy and a pursuit direction allocation method based on improved genetic algorithm are proposed, which can effectively achieve the pursuit task. The common problem with these algorithms ^17–22 is that the hunting time increases and the target escapes when some AUVs fail to reach the containment points at the same time.

In this article, the leader–follower formation algorithm is introduced into the multi-AUV hunting algorithm to solve the problems of long hunting time and target escaping in the hunting process. The AUVs that assign the same task are formed, and multiple AUVs keep the formation to track the target. Capture the target as the formation approaches. The specific process is as follows: firstly, the task is assigned according to the distance between the AUV and the target, and the nearest AUV forms a hunt team; then, the hunt team is formed in a V-shape and maintains the formation tracking target; and finally, when the formation is close to the target, the hunting point is determined according to the angle matching algorithm. Each AUV, respectively, reaches the corresponding hunting point to achieve the target capture. To prove the performance of the proposed algorithm, the target hunting was simulated in different scenarios. The simulation results show that the proposed algorithm can accomplish the hunting of moving targets. Compared with the BNN algorithm, the algorithm has higher success rate and shorter hunting time.

The main contributions of this article are (1) simulation in a 3-D underwater environments with underwater mountains, testing closer to the real environments; (2) the hunting strategy integrates algorithms such as leader–follower formation and angle matching to improve the hunting performance; and (3) through task assignment, formation tracking, and capturing targets, the capture time is shortened and the target escape is avoided.

The rest of the article is structured as follows: the “AUV kinematics model” section gives the model of the AUV; the algorithm design is given in the “Algorithm description” section; the “Simulation research” section gives the simulation test of the proposed algorithm; and finally, the conclusion and outlook for this article is given in the “Conclusion” section.

AUV kinematics model

To describe the navigation of the AUV, two coordinate systems are established: the body-fixed frame system {B} and the inertial frame system {E}. ²³ A 3-D Cartesian workspace is shown in Figure 1.

Figure 1.

Coordinate systems.

Inertial frame system

The inertial frame system takes a point E on the land or ocean as the origin. The EX axis and the EY axis are in the same horizontal plane, and the EY axis by the right-hand rule is EX axis clockwise rotate 90°. The EZ axis is perpendicular to the plane of the EX and EY axes, and the positive direction is the direction pointing to the center of the earth. The AUV position can be represented by the inertia coordinate $\left(x,y,z\right)$ of its center of gravity. The AUV position can also be expressed by the Euler angle $\left(\phi ,\theta ,\psi \right)$ of its carrier coordinate system relative to the inertial frame system, φ is the roll angle, θ is the pitch angle, and ψ is the yaw angle. Therefore, the state of the AUV can be represented by the vector $\eta ={[x y z \phi \theta \psi ]}^{\text{T}}$ .

Body-fixed frame system

The body-fixed frame system takes the center of gravity of AUV as the origin O, the vertical axis $O{X}_0$ is consistent with the main axis of symmetry of AUV, the direction of the stem is positive; the horizontal axis $O{Y}_0$ is parallel to the baseline plane, the starboard direction is positive; and the vertical axis $O{Z}_0$ is in the longitudinal section of AUV, which is perpendicular to the $X_{0}O{Y}_0$ plane and positive at the bottom direction. The motion state of the AUV in the body-fixed frame system is represented by a vector $V={[u v w p q r]}^{\text{T}}$ , u, v, w are the translational velocities of the AUV motion, p, q, r are the rotational velocities, and the corresponding force and moment are X, Y, Z, K, M, N.

Kinematic model

The state quantities of AUV such as velocity and acceleration are measured in the body-fixed frame system. The control vector of the AUV is generated by the motion controller, and the path and velocity of the AUV obtained in the inertial frame system need to be converted into the body-fixed frame system.

For the convenience of calculation, the position and angle information of the AUV are described in the inertial frame system, and the speed and angular velocity information are expressed in the carrier coordinate system, as described below ^24,25

${\eta }_1={\left[x,y,z\right]}^{\text{T}};{\eta }_2={\left[\phi ,\theta ,\psi \right]}^{\text{T}};\eta ={\left[{\eta }_1^{\text{T}},{\eta }_2^{\text{T}}\right]}^{\text{T}}$

$V_1={\left[u,v,w\right]}^{\text{T}};V_2={\left[p,q,r\right]}^{\text{T}};V={\left[V_1^{\text{T}},V_2^{\text{T}}\right]}^{\text{T}}$

${\tau }_1={\left[X,Y,Z\right]}^{\text{T}};{\tau }_2={\left[K,M,N\right]}^{\text{T}};\tau ={\left[{\tau }_1^{\text{T}},{\tau }_2^{\text{T}}\right]}^{\text{T}}$

The position under the inertial frame system and the velocity in the body-fixed frame system are transformed as follows

${\dot{\eta }}_1=J_1\left({\eta }_2\right)V_1$

1

Reverse conversion to get the speed value:

$V_1=J_1^{-1}\left({\eta }_2\right){\dot{\eta }}_1$

2

where

$\begin{array}{c}{J}_1({\eta }_2)=[\begin{array}{ccc}\text{cos}\psi \text{cos}\theta & -\text{sin}\psi \text{cos}\phi +\text{cos}\psi \text{sin}\theta \text{sin}\phi & \\ \text{sin}\psi \text{cos}\theta & \text{cos}\psi \text{cos}\phi +\text{sin}\psi \text{sin}\theta \text{sin}\phi & \\ -\text{sin}\theta & \text{cos}\theta \text{sin}\phi & \end{array}\\ \begin{array}{c}\text{sin}\psi \text{sin}\phi +\text{cos}\psi \text{cos}\phi \text{sin}\theta \\ -\text{cos}\psi \text{sin}\phi +\text{sin}\theta \text{sin}\psi \text{cos}\phi \\ \text{cos}\theta \text{cos}\phi \end{array}]\end{array}$

The angular velocity in the inertial frame system and the angular velocity in the body-fixed frame system are transformed as follows

${\dot{\eta }}_2=J_2\left({\eta }_2\right)V_2$

3

where

$J_2\left({\eta }_2\right)=\left[\begin{array}{ccc}1& \text{sin}\phi \text{tan}\theta & \text{cos}\phi \text{tan}\theta \\ 0& \text{cos}\phi & -\text{sin}\phi \\ 0& \text{sin}\phi /\text{cos}\theta & \text{cos}\phi /\text{cos}\theta \end{array}\right]$

In summary, the AUV kinematics equation is obtained, as shown in equation (4)

$\left[\begin{array}{l}{\dot{\eta }}_1\\ {\dot{\eta }}_2\end{array}\right]=\left[\begin{array}{cc}{J}_1\left({\eta }_2\right)& 0_{3\times 3}\\ 0_{3\times 3}& J_1\left({\eta }_2\right)\end{array}\right]\left[\begin{array}{l}{V}_1\\ V_2\end{array}\right]\iff \dot{\eta }=J\left(\eta \right)V$

4

Dynamic model

Typically, an AUV is usually required to move close to the deep-sea bottom with complex terrain. When it moves to resist the ocean current in deep sea, the AUV’s motion status will be in large drift angles. When it is required to move at a low forward/rotational speed, the nonlinear characteristics are strong with maneuverability hydrodynamic forces of the AUV. It is necessary to make special mathematical model for some situations. To give a more clear explanation for the latter control approach, the dynamic equation of AUV can be presented as a compact vector form ²⁶

$M\dot{q}+C\left(q\right)q+D\left(q\right)q+g\left(\eta \right)=\tau$

5

where $M\in R^{n\times n}$ is the inertia matrix including the added mass effects; $C\in R^{n\times n}$ is the matrix of Coriolis and centrifugal terms (including added mass caused by hydrodynamic effect); $D\in R^{n\times n}$ is the hydrodynamic damping matrix; $g\in R^n$ is the gravity and buoyancy forces; and $\tau \in R^n$ is the control forces and moments acting on the AUV center of mass.

As mentioned earlier, the system dynamics are not exactly known. The system dynamics can be divided into two parts: estimated dynamics $\widehat{\tau }$ and unknown dynamics $\stackrel{\sim }{\tau }$

$\tau =\widehat{\tau }+\stackrel{\sim }{\tau }$

6

where $\widehat{\tau }=\widehat{M}\dot{q}+\widehat{C}q+\widehat{D}q+\widehat{g}$ , $\stackrel{\sim }{\tau }=\stackrel{\sim }{M}\dot{q}+\stackrel{\sim }{C}q+\stackrel{\sim }{D}q+\stackrel{\sim }{g}+{\tau }_d$ , and $\widehat{M},\widehat{C},\widehat{D},\widehat{g}$ are estimated terms, $\stackrel{\sim }{M},\stackrel{\sim }{C},\stackrel{\sim }{D},\stackrel{\sim }{g}$ are the unknown terms, and ${\tau }_d$ is the disturbance term.

Algorithm description

In this article, m AUVs in a 3-D underwater environments are used to hunt n targets. The whole workflow is as follows: (1) After any AUV finds the target, the system will assign tasks according to the distance between all AUVs and the target; (2) the AUV that discovers the target is defined as the leader, and the remaining AUVs for the same task are defined as followers. Leaders and followers are organized in a leader–follower mode and keep the formation to track the target; and (3) when the AUV formation is close to the target, the hunting point is determined according to the angle matching method. AUVs reach the hunting point to round up the target. The entire workflow is shown in Figure 2.

Figure 2.

Hunting task flow chart.

To complete multi-AUV collaborative target hunting in a 3-D underwater environments, some key issues need to be addressed, including task assignment, leader–follower formation, and target capture. The specific algorithms of these three parts are described below.

Task assignment

When the target is discovered, which AUVs participate in the hunting is determined by the task assignment algorithm. The algorithm assigns tasks according to the distance between the AUV and the target to form a hunting team. In this article, it is assumed that one target requires six AUVs to be hunted in a 3-D underwater environments. Assume that the position of the ith target is $S_i$ . $R_{\text{desired}}$ is the closest AUV to the target, which can be expressed by the relationship of (7) and (8) ²⁷

$R_{\text{desired}}\Leftarrow P_{\text{desired}}=\text{min}\left\{D_{ij}\right\} \text{for} R_{\text{desired}}\in \Omega$

7

$D_{ij}=|S_i-P_j|=\sqrt{{(s_{ix}-p_{jx})}^2+{(s_{iy}-p_{jy})}^2+{(s_{iz}-p_{jz})}^2}$

8

where $D_{ij}$ is the Euclidean distance between the j-th AUV and the ith target. $S_i=(s_{ix},s_{iy},s_{iz})$ represents the position coordinate of the ith target and $P_j=(p_{jx},p_{jy},p_{jz})$ represents the position coordinate of the j-th AUV. $P_{\text{desired}}$ is the closest distance between the AUV and the target $S_i$ . Ω is the set of all AUVs. Repeat the above selection process to select the nearest six AUVs.

Formation tracking

After the task assignment is completed, the AUVs that are assigned the same task will be teamed to track the target. The AUV that discovers the target as the leader, and the remaining AUVs in the hunting team as the follower. Firstly, a virtual AUV is designed according to the speed and position of the leader AUV. Then, the backstep control strategy is introduced to design the formation control rate for the AUV. Finally, the virtual AUV is tracked by the follower AUV to achieve the expected formation effect. The navigation trajectory of the virtual AUV in the inertial frame system is ${\eta }_v={[x_v y_v z_v {\psi }_v]}^{\text{T}}$ , and speed information $V_v={[u_v v_v w_v r_v]}^{\text{T}}$ , where $(x_v,y_v,z_v)$ is the position information of the virtual AUV in the inertial frame system and ${\psi }_v$ is the angle value of the AUV rotating counterclockwise around the Z-axis. The actual navigational state of follower AUV is defined as ${\eta }_f={[x_f y_f z_f {\psi }_f]}^{\text{T}}$ . To realize the formation control task and make the AUV sail according to the desired formation, it is necessary to control the speed and angular velocity $V_f={[u_f v_f w_f r_f]}^T$ of the following AUV so that it can track the trajectory of the virtual AUV. Finally, the formation control error e_e converges to 0 in the inertial frame system ²⁸

$e_e={\eta }_v-{\eta }_f=\left[\begin{array}{l}{e}_x\\ e_y\\ e_z\\ e_{\psi }\end{array}\right]=\left[\begin{array}{l}{x}_v-x_f\\ y_v-y_f\\ z_v-z_f\\ {\psi }_v-{\psi }_f\end{array}\right]$

9

The structural block diagram of the control system is shown in Figure 3.

Figure 3.

Basic structure block diagram of kinematic formation control system.

In the kinematic formation control system of Figure 3, the virtual AUV speed information $V_v={[u_v v_v w_v r_v]}^{\text{T}}$ and the position information ${\eta }_v={[x_v y_v z_v {\psi }_v]}^{\text{T}}$ are obtained using the leader AUV position information ${\eta }_m={[x_m y_m z_m {\psi }_m]}^{\text{T}}$ . The error between leader AUV and virtual AUV trajectory and speed of virtual AUV are used as the input of backstep control. The kinematic formation control law is obtained by the backstep method ^29,30

$\begin{array}{l}{V}_f=\left[\begin{array}{l}{u}_f\\ v_f\\ w_f\\ r_f\end{array}\right]=J^{\text{T}}({\eta }_f)K{e}_e+J(e_{\psi })V_v\\ =\left[\begin{array}{l} \text{cos}{\psi }_f \text{sin}{\psi }_f 0 0\\ -\text{sin}{\psi }_f \text{cos}{\psi }_f 0 0\\ 0 0 1 0\\ 0 0 0 1\end{array}\right]\cdot \left[\begin{array}{l}k\\ k\\ k_z\\ k_{\psi }\end{array}\right]\left[\begin{array}{l}{e}_x\\ e_y\\ e_z\\ e_{\psi }\end{array}\right]\text{+}\left[\begin{array}{l}\text{cos}{e}_{\psi } -\text{sin}{e}_{\psi } 0 0\\ \text{sin}{e}_{\psi } \text{cos}{e}_{\psi } 0 0\\ 0 0 1 0\\ 0 0 0 1\end{array}\right]\cdot \left[\begin{array}{l}{u}_v\\ v_v\\ w_v\\ r_v\end{array}\right]\\ =\left[\begin{array}{c}k(e_x\text{cos}\hspace{0.17em}{\psi }_f+e_y\hspace{0.17em}\text{sin}\hspace{0.17em}{\psi }_f)+(u_v\hspace{0.17em}\text{cos}\hspace{0.17em}{e}_{\psi }-v_v\hspace{0.17em}\text{sin}\hspace{0.17em}{e}_{\psi })\\ k(-e_x\hspace{0.17em}\text{sin}\hspace{0.17em}{\psi }_f+e_y\hspace{0.17em}\text{cos}\hspace{0.17em}{\psi }_f)+(u_v\hspace{0.17em}\text{sin}\hspace{0.17em}{e}_{\psi }+v_v\hspace{0.17em}\text{cos}\hspace{0.17em}{e}_{\psi })\\ w_v+k_z{e}_z\\ r_v+k_{\psi }{e}_{\psi }\end{array}\right]\end{array}$

10

where k, k_z , k_ψ are positive constant coefficients.

Target capture

After the multi-AUV formation is close to the target, it will form an encirclement as soon as possible to achieve the capture of the target. When the target capture is started, the position of AUV in the formation is $P_j(j=\text{1},\text{2},\mathrm{...},\text{6)}$ ; the position of the ith target is $S_i$ . $\overrightarrow{S_i{P}_j}$ represents the direction vector of the target to AUV in formation. An angle matching algorithm is introduced to determine the hunting point. Each AUV follows the steps below to obtain the ideal hunting point position. ^31,32

Step 1

Set the polar coordinates ${\displaystyle \sum {}_{S_i}}$ with $S_i$ as the pole and $\overrightarrow{S_i{P}_j}$ as the polar axis.

Step 2

Calculate the coordinates $P_s^{mi}({\lambda }_m,{\phi }_m) (m=\text{1},\mathrm{...},\text{4})$ of all AUVs in the formation in polar coordinates ${\displaystyle \sum {}_{S_i}}$ , where ${\phi }_m\in [\text{0},\text{2π})$ .

Step 3

Generate the hunting points for all AUVs in the formation. The hunting points are four points evenly distributed on a circle with $S_i$ as the center and $r_{sc}$ as the radius. The hunting point are defined as $P_d^{mi}(r_{sc},{\phi }_n)(n=\text{1},\mathrm{...},\text{4})$ in the polar coordinates ${\displaystyle \sum {}_{S_i}}$ , where ${\phi }_n=\frac{\text{2π}}{N}n$ . Correspondingly, the coordinates of the hunting point in the inertial frame system {E} are $P_W^i(x_n,y_n)(n=\text{1},\text{2},\mathrm{...},\text{4})$ .

Step 4

The AUV reaches the hunting point along the shortest path. Here is a simple and effective angle strategy. AUV chooses the hunting point according to the principle that ${\phi }_m$ is larger and ${\phi }_n$ is larger.

Remark

When the target is successfully hunting, the positions of AUVs are shown in Figure 4. Since the positions of AUV R ₁ and R ₆ are directly above and directly below the target, the angle matching algorithm is only needed to determine the positions of the remaining four hunting points in the same plane.

Figure 4.

Positions of the AUVs when successfully hunting the target. AUV: autonomous underwater vehicle.

Simulation research

To test the feasibility and effectiveness of the proposed algorithm, a simulation experiment is given based on the MATLAB platform. To more clearly reflect the process of target hunting, the simulation is divided into two parts: formation and target hunting.

Formation control simulation

7AUV linear navigation formation control simulation

In this section, the 7AUV linear navigation formation control simulation is performed. The navigation trajectory of the leader AUV is set to be linear: ${\eta }_m={[x_m y_m z_m {\psi }_m]}^{\text{T}}$ , where $x_m=0.3t+20$ ; $y_m=10+0.3t$ ; $z_m=0.3t$ ; ${\psi }_m=\pi /4$ . The six follower AUV are $R_1, R_2,R_3, R_4,R_5, R_6$ . The simulation environments are set to the 3-D environments of 30 × 30 × 25 m³. $R_1,R_2$ follows $R_0$ , $R_4$ follows $R_2$ , $R_6$ follows $R_4$ , $R_3$ follows $R_1$ , $R_5$ follows $R_3$ . For $R_0, R_1, R_2,R_3, R_4,R_5, R_6$ , the formation parameters are presented in Table 1.

Table 1.

7AUV linear navigation formation parameters.

AUV	Parameter	Value
$\begin{array}{c}{R}_0, R_1, R_2, R_3,R_4,R_5, R_6\end{array}$	d	4 m
	α	$\pi /4$
	${\lambda }_1$	2
	${\lambda }_2$	1
	${\lambda }_3$	4
	${\lambda }_4$	0.5
	$k_1$	2
	$k_2$	1
	$k_3$	2
	$k_4$	0.1
$R_0, R_2, R_4, R_6$	β	$-2\pi /3$
$R_1,R_3,R_5$	β	$2\pi /3$

AUV: autonomous underwater vehicle.

The kinematic control parameters are set to be $k=1$ , $k_z=2$ , $k_{\psi }=2$ . The initial position of R ₀ is (20 10 0 0.79); the initial position of R ₁ is (15 10 −2.83 0); the initial position of R ₂ is (20 5 −2.82 0); the initial position of R ₃ is (17 10 −5.66 0); the initial position of R ₄ is (20 2 −5.66 0); the initial position of R ₅ is (10 10 −8.49 0); and the initial position of R ₆ is (20 0 −8.49 0). Set the simulation time to 50 s, and the virtual AUV initial position value is set according to the corresponding initial value of the desired trajectory.

The kinematics formation control law of follower AUVs is obtained by the backstep formation control method. The formation process of 7AUV is shown in Figure 5. AUV $R_0$ is the leader, AUV $R_1, R_2, R_3, R_4,R_5,R_6$ are the followers. At the initial moment of the simulation, there is a certain error between the AUV actual formation and the expected formation control formation. Since the initial position of the AUV is arbitrarily set, the following AUV can adjust its position in real time according to the kinematic formation control law, gradually form the desired formation control formation, and maintain the desired formation during the AUV navigation. It can be seen from the simulation of the formation control that the methods mentioned in this article have realized the linear navigation V-type formation control.

Figure 5.

7AUV linear formation control. AUV: autonomous underwater vehicle.

7AUV spiral navigation formation control simulation

In the simulation of spiral navigation formation control, the position information of the leader AUV is first given. Set the navigation path of the leader AUV R ₀ to a spiral: ${\eta }_m={[x_m y_m z_m {\psi }_m]}^{\text{T}}$ , where $x_m=5\text{sin}(0.2t)$ ; $y_m=-5\text{cos}(0.2t)$ ; $z_m=0.15t$ ; ${\psi }_m=0.2t$ . The six follower AUVs are $R_1, R_2, R_3, R_4, R_5,R_6$ . The 3-D simulation environments are set to 16 × 16 × 11 m³. $R_1,R_2$ follows R ₀, R₄ follows R ₂, R₆ follows R ₄, R₃ follows R ₁, and R ₅ follows R ₃. For $R_0,R_1, R_2,R_3,R_4, R_5,R_6$ , the formation parameters are shown in Table 2.

Table 2.

7AUV spiral navigation formation parameters.

AUV	Parameter	Value
$\begin{array}{l}{R}_0, R_1, R_2, R_3,R_4, R_5, R_6\end{array}$	d	1.5 m
	α	$\pi /4$
	${\lambda }_1$	2
	${\lambda }_2$	1
	${\lambda }_3$	4
	${\lambda }_4$	0.5
	$k_1$	2
	$k_2$	1
	$k_3$	2
	$k_4$	0.1
$R_0, R_2, R_4, R_6$	β	$-2\pi /3$
$R_1,R_3,R_5$	β	$2\pi /3$

AUV: autonomous underwater vehicle.

The kinematic control parameters are set to be $k=1$ , $k_z=2$ , and $k_{\psi }=2$ . The initial position of R₀ is (0 −5 0 0); the initial position of R₁ is (−1 −4 −1 0); the initial position of R₂ is (−1 −6 −1.06 0); the initial position of R₃ is (0 −3 −2.12 0); the initial position of R₄ is (−1 −7 −2.12 0); the initial position of R ₅ is (−1 −2 −3.18 0); and the initial position of R ₆ is (−2 −8 −3.18 0). Set the simulation time to 50 s, and the initial position value of the virtual AUV is set according to the corresponding initial value of the desired trajectory. The simulation result is shown in Figure 6. In Figure 6, numbers 1 and 2 AUV follows the leader AUV (number 0), 4 follows 2, 6 follows 4, 3 follows 1, and 5 follows 3, the navigation information of the leader AUV decide the whole formation, and combining with the following and tracking process above, the spiral formation control is realized.

Figure 6.

7AUV spiral formation control. AUV: autonomous underwater vehicle.

Target hunting simulation

In this section, the entire target hunting process is simulated. To simplify the implementation, the following assumptions are made in this study: (1) the AUV and the target are assumed to have no shape at the particle point, and the direction of their motion can be changed without any delay; (2) the target has certain intelligence and can avoid the capture of AUV; (3) the speed of the AUV is faster than the target; and (4) xix AUVs required to round up a target.

Remark

The shape of the AUV is an important factor that must be considered in actual control. To solve this problem, size of obstacles are appropriately increased in the simulation.

Single target hunting

First, simulate a situation where only one target needs to be hunting. At the beginning of the hunting, the distribution of the seven AUVs and the target position is shown in Figure 7(a). Suppose that R₁ finds the target T at this time. The capture mission begins, the task is assigned first. Because R₇ is furthest away from the target, it is not assigned a task. The hunting mission is completed by $R_{\text{1}},\mathrm{...},R_{\text{6}}$ . After the task assignment is completed, R₁ tracks the T as the leader, $R_{\text{2}},\mathrm{...},R_{\text{6}}$ starts the formation as a follower. Since the target has intelligent, it will evade in the direction of no AUV, as shown in Figure 7(b). As shown in Figure 7(c), when the formation is completed, the AUV system will determine the hunting points of each AUV in the formation according to the angle matching algorithm. The AUV will arrive at the nearest hunting point along the straight-line to complete the capture of the target. Figure 7(d) shows the entire target hunting process. From the simulation results, the proposed algorithm can effectively accomplish the single target hunting in the 3-D underwater environments.

Figure 7.

Single target hunting simulation: (a) initial position, (b) task assignment, (c) formation tracking, and (d) complete hunt.

Multiple target hunting

In the actual hunting mission, often more than one target exists. The simulation of multiple targets hunting is shown in Figure 8. The 10 AUVs in the simulation need to capture two targets. As shown in Figure 8(a), the distribution of AUVs and targets is given. When AUV R₁ and R₄ find target T₁ and T₂, respectively, the hunting task is assigned. As the number of AUVs does not meet the requirements for simultaneous hunting of two targets. Therefore, only through task assignment, one target can be successfully hunted and then another target can be hunted. According to the task assignment principle, AUV R₃, R₄, R₇, R₈, R₉, and $R_{\text{10}}$ are assigned to T₁, while AUVs R₁, R₂, R₅, and R₆ are assigned to target T₂. Next, six AUV formations and hunting the target T₁. Unable to hunt target T₂, AUVs R₁, R₂, R₅, and R₆ will track the formation to track the target T₂. Figure 8(b) shows the situation where the first target is captured. At the same time, it can be seen that AUV R₁, R₂, R₅, and R₆ are in a V-shaped formation to track the target T₂. After the T₁ is captured, the mission is assigned again. AUV R₇ and R₉ will continue to participate in the hunting of T₂. The target T₂ is captured when R₇ and R₉ complete the new formation with the original R₁, R₂, R₅, and R₆. The result is shown in Figure 8(c). The simulation results show that the AUV system based on the proposed algorithm has good cooperation and can complete the hunting of multiple targets in time.

Figure 8.

Multiple target hunting simulation: (a) initial position, (b) the first target is hunted, (c) the second target is hunted.

Discussion

The results of the simulation experiments in the “Formation control simulation” and “Target hunting simulation” sections show that the proposed approach can satisfy the cooperative hunting task by multiple AUVs under 3-D underwater environments. The effects of the main design parameters on the control performances are discussed in this section. We select main design parameters k and $k_{\psi }$ for discussion. k and $k_{\psi }$ are positive constant coefficients, and their values affect the hunting control accuracy.

To analyze the influence of parameters k and $k_{\psi }$ , some simulations are carried out under the same conditions as the first simulation experiment in the “Target hunting simulation” section, except choosing different values of parameters k and $k_{\psi }$ . Because a lot of things are random, every simulation is conducted 10 times. The error curves of the AUV R ₁ at k = 5, k = 50, and k = 100 are shown in Figure 9. The error curves of the AUV R ₁ at $k_{\psi }=1$ , $k_{\psi }=5$ , and $k_{\psi }=10$ are shown in Figure 10.

Figure 9.

Error curves for different k values.

Figure 10.

Error curves for different $k_{\psi }$ values.

It can be seen from the simulation comparison results that the hunting performance of the AUV is improved by selecting larger parameters k and $k_{\psi }$ . But performance has not improved much.

The results in Figures 9 and 10 show that the proposed approach is not very sensitive to the variations of the model parameters. (The difference between the error curves of different values is small.) The parameters can be chosen in a very wide range. Therefore, different values of main design parameters have little influence on the control performances.

Comparison of different algorithms

The simulation results of the first two sections show that the algorithm mentioned in this article can complete the multi-AUV target hunting in the 3-D unknown environments. The performance of the algorithm is discussed in this section. To verify the success rate of the proposed algorithm, it is compared with the BNN algorithm. The BNN algorithm sets the target as an external excitation signal and directs it to the AUV to the target through the transmission of the neural network. At the same time, the obstacle is set as an external suppression signal, and the AUV is partially rejected to avoid collision with obstacles. ³³ Assume that in the 3-D environments of 100 × 100 ×60 m³, six AUVs hunt one target. Each time the position of the AUVs and the target are randomly distributed, 100 simulations are passed. The comparison results of the two algorithms are presented in Table 3. Table 3 lists the hunting success rates of the two algorithms, the average distance traveled by each AUV, and the average hunting time.

Table 3.

Comparison of performance of two algorithms.

	Proposed Algorithm	BNN
Hunting success rate (%)	100	89
Average hunting time (s)	36.52	68.78
Average distance traveled by each AUV
R 1 (m)	68.32	87.63
R 2 (m)	84.53	98.07
R 3 (m)	78.15	105.37
R 4 (m)	67.32	93.56
R 5 (m)	79.25	83.78
R 6 (m)	82.94	102.73

AUV: autonomous underwater vehicle.

It can be seen from the table that the success rate of the proposed algorithm is higher than the success rate of the BNN algorithm, the hunting time is short, and the average distance of each AUV navigation is shorter. Analyze the reason:

The BNN algorithm was used to plan the hunting path, each AUV moves toward the direction with the largest activity value of the neuron, which is easy to cause conflict, resulting in an increase in navigation distance and a decrease in hunting efficiency. In the proposed algorithm, multi-AUV do not cause conflicts because the system is in formation while hunting the target.

In the BNN algorithm, AUVs do not reach the hunting point at the same time, which easily causes the target to escape. In the proposed algorithm, the system is formed, and the all AUVs can reach the hunting points at the same time to avoid the escape of the target.

The BNN algorithm needs to calculate the activity value of each neuron for each iteration, and the calculation amount is large, resulting in slow hunting speed.

Conclusion

This article studies the multi-AUV collaborative target hunting in the 3-D underwater environments and proposes a target hunting method based on leader–follower formation. The mentioned algorithm improves the efficiency of hunting. The hunting of single or multiple targets under water is achieved through three stages: task assignment, formation tracking, and target capture. With the introduction of leader–follower formation, all AUVs arrive at the hunting points at the same time, which shortens the hunting time and avoids the target escaping. Compared with the BNN algorithm, the proposed algorithm not only has better performance in hunting efficiency and hunting time, but also improves the success rate of hunting. Although the algorithm mentioned herein has the above advantages, there are still some deficiencies. Future work will further improve existing algorithms to provide better performance. At the same time, the environment and AUV models will be further explored to enable the algorithm to have a wider range of applications.